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Lajos Diósi

A Short Course in QuantumInformation Theory

An Approach From Theoretical Physics

Second Edition

123

Lajos DiósiMTA BudapestKFKI Research Institute for Particle and

Nuclear Physics (RMKI)Konkoly Thege Miklós út 29-331525 BudapestHungarye-mail: diosi@rmki.kfki.hu

ISSN 0075-8450 e-ISSN 1616-6361

ISBN 978-3-642-16116-2 e-ISBN 978-3-642-16117-9

DOI 10.1007/978-3-642-16117-9

Springer Heidelberg Dordrecht London New York

� Springer-Verlag Berlin Heidelberg 2011

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Preface

Quantum information has become an independent fast growing research field.There are new departments and labs all around the world, devoted to particular oreven complex studies of mathematics, physics, and technology of controllingquantum degrees of freedom. The promised advantage of quantum technologieshas obviously electrified the field which had been considered a bit marginal untilquite recently. Before, many foundational quantum features had never been testedor used on single quantum systems but on ensembles of them. Illustrations ofreduction, decay, or recurrence of quantum superposition on single states went tothe pages of regular text-books, without ever being experimentally tested. Now-adays, however, a youngest generation of specialists has imbibed quantum theo-retical and experimental foundations ‘‘from infancy’’.

From 2001 on, in spring semesters I gave special courses for under- andpostgraduate physicists at Eötvös University. The twelve lectures could not includeall standard chapters of quantum information. My guiding principles were those ofthe theoretical physicist and the believer in the unity of physics. I achieved adecent balance between the core text of quantum information and the chapters thatlink it to the edifice of theoretical physics. Scholarly experience of the past fivesemesters will be utilized in this book

I suggest this thin book for all physicists, mathematicians and other peopleinterested in universal and integrating aspects of physics. The text does not requirespecial mathematics but the elements of complex vector space and of probabilitytheories. People with prior studies in basic quantum mechanics make the perfectreaders. For those who are prepared to spend many times more hours with quantuminformation studies, there are exhaustive monographs written by Preskill, byNielsen and Chuang, or the edited one by Bouwmeester, Ekert, and Zeilinger. Andfor each of my readers, it is almost compulsory to find and read a second thin book‘‘Short Course in Quantum Information, approach from experiments’’. . .

Acknowledgements I benefited from conversations and/or correspondence withJürgen Audretsch, András Bodor, Todd Brun, Tova Feldmann, Tamás Geszti,Thomas Konrad, and Tamás Kiss. I am grateful to them all for the generous helpand useful remarks that served to improve my manuscript.

v

It is a pleasure to acknowledge financial support from the Hungarian ScientificResearch Fund, Grant No. 49384.

Budapest, February 2006 Dr. Lajos Diósi

vi Preface

Preface (extended 2nd edition)

Following the publisher’s suggestion, I prepared this extended 2nd edition toinclude new parts and corrections to the 1st one. I also felt encouraged by myexperience of continued teaching this course at Eötvös University, and of teachingspecial courses at Technion and at Durban University. The structure and contentsof the volume have not changed much. However, a 12th chapter on ‘‘Qubitthermodynamics’’ and a related Appendix have been added. There are new sec-tions on ‘‘Weak measurement, time-continuous measurement’’ in both Chaps. 2and 4, and a summary on ‘‘Fock representation’’ of qubits in Chap. 5. Two newsections were added to Chap. 11, the one on ‘‘Period finding quantum algorithm’’and the other on ‘‘Quantum error correction’’. A handful of new figures, visual-izing the common ‘‘urn model’’ of statistics, intend to serve the reader’s conve-nience. The 2nd edition gives me an opportunity to eliminate (hopefully most)errors or deficiencies in the 1st edition. This concerns basically Sects. 8.5 and 11.5,Sol. 4.1 in the 1st edition. Additional references bring into the reader’s attentionsome recently published textbooks. The present volume remains a special one forit builds on the links between physics foundations and quantum information, andfor its moderate length.

Acknowledgements I’m indebted to Ady Mann for his comments and carefulreading of Chaps. 1–5, and to Tamás Geszti and Michael Revzen for their usefulremarks. It is a pleasure to acknowledge financial support from the HungarianScientific Research Fund, Grant No. 75129.

Budapest, May 2010 Dr. Lajos Diósi

vii

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Foundations of Classical Physics . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 State Space, Equation of Motion . . . . . . . . . . . . . . . . . . . . . 72.2 Operation, Mixing, Selection . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Linearity of Non-Selective Operations. . . . . . . . . . . . . . . . . . 92.4 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4.1 Projective Measurement . . . . . . . . . . . . . . . . . . . . . . 102.4.2 Non-Projective Measurement . . . . . . . . . . . . . . . . . . . 122.4.3 Weak Measurement, Time-Continuous Measurement. . . 12

2.5 Composite Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.6 Collective System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.7 Two-State System (Bit) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.8 Problems, Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Semiclassical, Semi-Q-Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.1 Problems, Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Foundations of Q-Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.1 State Space, Superposition, Equation of Motion . . . . . . . . . . . 234.2 Operation, Mixing, Selection . . . . . . . . . . . . . . . . . . . . . . . . 254.3 Linearity of Non-Selective Operations. . . . . . . . . . . . . . . . . . 264.4 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.4.1 Projective Measurement . . . . . . . . . . . . . . . . . . . . . . 274.4.2 Non-Projective Measurement . . . . . . . . . . . . . . . . . . . 294.4.3 Weak Measurement, Time-Continuous Measurement. . . 30

ix

4.4.4 Compatible Physical Quantities . . . . . . . . . . . . . . . . . 314.4.5 Measurement in Pure State . . . . . . . . . . . . . . . . . . . . 32

4.5 Composite Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.6 Collective System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.6.1 Problems, Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 36References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5 Two-State Q-System: Qubit Representations . . . . . . . . . . . . . . . . 375.1 Computational Representation . . . . . . . . . . . . . . . . . . . . . . . 375.2 Pauli Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.2.1 State Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.2.2 Rotational Invariance . . . . . . . . . . . . . . . . . . . . . . . . 405.2.3 Density Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.2.4 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . 415.2.5 Physical Quantities, Measurement . . . . . . . . . . . . . . . 42

5.3 The Unknown Qubit, Alice and Bob. . . . . . . . . . . . . . . . . . . 435.4 Relationship of Computational and Pauli Representations . . . . 435.5 Fock Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.6 Problems, Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6 One-Qubit Manipulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.1 One-Qubit Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.1.1 Logical Operations . . . . . . . . . . . . . . . . . . . . . . . . . . 476.1.2 Depolarization, Re-Polarization, Reflection . . . . . . . . . 49

6.2 State Preparation, Determination . . . . . . . . . . . . . . . . . . . . . 506.2.1 Preparation of Known State, Mixing. . . . . . . . . . . . . . 506.2.2 Ensemble Determination of Unknown State. . . . . . . . . 516.2.3 Single State Determination: No-Cloning . . . . . . . . . . . 526.2.4 Fidelity of Two States . . . . . . . . . . . . . . . . . . . . . . . 536.2.5 Approximate State Determination and Cloning . . . . . . 53

6.3 Indistinguishability of Two Non-Orthogonal States . . . . . . . . . 546.3.1 Distinguishing Via Projective Measurement. . . . . . . . . 546.3.2 Distinguishing Via Non-Projective Measurement . . . . . 55

6.4 Applications of No-Cloning and Indistinguishability . . . . . . . . 566.4.1 Q-banknote . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.4.2 Q-key, Q-Cryptography. . . . . . . . . . . . . . . . . . . . . . . 57

6.5 Problems, Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7 Composite Q-System, Pure State . . . . . . . . . . . . . . . . . . . . . . . . . 617.1 Bipartite Composite Systems . . . . . . . . . . . . . . . . . . . . . . . . 61

7.1.1 Schmidt Decomposition . . . . . . . . . . . . . . . . . . . . . . 627.1.2 State Purification . . . . . . . . . . . . . . . . . . . . . . . . . . . 627.1.3 Measure of Entanglement . . . . . . . . . . . . . . . . . . . . . 63

x Contents

7.1.4 Entanglement and Local Operations . . . . . . . . . . . . . . 657.1.5 Entanglement of Two-Qubit Pure States . . . . . . . . . . . 667.1.6 Interchangeability of Maximal Entanglements . . . . . . . 67

7.2 Q-Correlations History . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687.2.1 EPR, Einstein Nonlocality 1935 . . . . . . . . . . . . . . . . . 687.2.2 A Non-Existing Linear Operation 1955. . . . . . . . . . . . 697.2.3 Bell Nonlocality 1964. . . . . . . . . . . . . . . . . . . . . . . . 70

7.3 Applications of Q-Correlations . . . . . . . . . . . . . . . . . . . . . . . 737.3.1 Superdense Coding . . . . . . . . . . . . . . . . . . . . . . . . . . 737.3.2 Teleportation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

7.4 Problems, Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

8 All Q-Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798.1 Completely Positive Maps . . . . . . . . . . . . . . . . . . . . . . . . . . 798.2 Reduced Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 808.3 Indirect Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818.4 Non-Projective Measurement Resulting from Indirect

Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 838.5 Entanglement and LOCC. . . . . . . . . . . . . . . . . . . . . . . . . . . 848.6 Open Q-System: Master Equation. . . . . . . . . . . . . . . . . . . . . 858.7 Q-Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 858.8 Problems, Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

9 Classical Information Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 899.1 Shannon Entropy, Mathematical Properties . . . . . . . . . . . . . . 899.2 Messages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 909.3 Data Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 909.4 Mutual Information. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 929.5 Channel Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 939.6 Optimal Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 949.7 Cryptography and Information Theory . . . . . . . . . . . . . . . . . 949.8 Entropically Irreversible Operations . . . . . . . . . . . . . . . . . . . 959.9 Problems, Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

10 Q-Information Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9710.1 Von Neumann Entropy, Mathematical Properties . . . . . . . . . . 9710.2 Messages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9810.3 Data Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9910.4 Accessible Q-Information . . . . . . . . . . . . . . . . . . . . . . . . . . 10110.5 Entanglement: The Resource of Q-Communication. . . . . . . . . 10210.6 Entanglement Concentration (Distillation) . . . . . . . . . . . . . . . 103

Contents xi

10.7 Entanglement Dilution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10410.8 Entropically Irreversible Operations . . . . . . . . . . . . . . . . . . . 10510.9 Problems, Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

11 Q-Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10911.1 Parallel Q-Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10911.2 Evaluation of Arithmetic Functions. . . . . . . . . . . . . . . . . . . . 11011.3 Oracle Problem: The First Q-Algorithm . . . . . . . . . . . . . . . . 11111.4 Searching Q-Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11311.5 Fourier Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11511.6 Period Finding Q-Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 11611.7 Error Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11811.8 Q-Gates, Q-Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12011.9 Problems, Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

12 Qubit Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12312.1 Thermal Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12312.2 Ideal Qubit Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12412.3 Informatic and Thermodynamic Entropies . . . . . . . . . . . . . . . 12512.4 Q-Thermalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12612.5 Q-Refrigerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12712.6 Thermal Qubit with External Work. . . . . . . . . . . . . . . . . . . . 12912.7 Q-Carnot Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13012.8 Problems, Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

xii Contents

Symbols, Acronyms, Abbreviations

{ , } Poisson bracket[ , ] Commutator{ , } Anti-commutatorh i Expectation value

O MatrixOy Adjoint matrixx, y, . . . Phase space pointsC Phase spaceq(x) Phase space distribution, Classical statex, y, . . . Binary numbersx1x2 . . . xn Binary stringq(x) Discrete classical stateM OperationT Polarization reflectionI Identity operationL Lindblad generatorA(x), A(x) Classical physical quantitiesH(x) Hamilton functionP Indicator functionPðxÞ;PðxÞ Classical effectH Hilbert spaced Vector space dimensionjwi; jui; . . . State vectorshwj; huj; . . . Adjoint state vectorshwjui Complex inner producthwjOjui Matrix elementq Density matrix, Quantum stateA Quantum physical quantityH HamiltonianP Hermitian projector

xiii

I Unit matrixÛ Unitary mapP Quantum effectp Probabilityw Weight in mixturejxi Computational basis vectorx Qubit Hermitian matrixq- QuantumcNOT Controlled NOT� Modulo sum� Composition9 Cartesian product� Tensor producttr TracetrA Partial tracej"i; j#i Spin-up, spin-down basisn, m . . . Bloch unit vectorsjni Qubit state vectors Qubit polarization vectorrx; ry; rz Pauli matricesr Vector of Pauli matricesa, b, a, … Real spatial vectorsab Real scalar productâ, â� Emission, absorption matricesX, Y, Z One qubit Pauli gatesH Hadamard gateT(u) Phase gateF FidelityE Entanglement measureS(q), S(p) Shannon entropySðqÞ von Neumann entropySðq0jjqÞ; Sðq0jjqÞ Relative entropyjW�i; jU�i Bell basis vectorsMn Kraus matrices|n; Ei Environmental basis vectorX, Y, . . . Classical messageH(X), H(Y) Shannon entropyH(X|Y) Conditional Shannon entropyI(X:Y) Mutual informationC Channel capacityq(x|y) Conditional stateq(y|x) Transfer functionT Temperature

xiv Symbols, Acronyms, Abbreviations

E EnergyW WorkQ HeatSth Thermodynamic entropyLO Local OperationLOCC Local Operation and Classical Communication

Symbols, Acronyms, Abbreviations xv

Chapter 1Introduction

1.1 Introduction

Classical physics—the contrary to quantum—means all those fundamentaldynamical phenomena and their theories which became known until the end of thenineteenth century, from our studying the macroscopic world. Galileo’s, Newton’s,and Maxwell’s consecutive achievements, built one on top of the other, obtainedtheir most compact formulation in terms of the classical canonical dynamics. Atthe same time, the conjecture of the atomic structure of the microworld was alsoconceived. By extending the classical dynamics to atomic degrees of freedom,certain microscopic phenomena also appearing at the macroscopic level could beexplained correctly. This yielded indirect, yet sufficient, proof of the atomicstructure. But other phenomena of the microworld (e.g., the spectral lines ofatoms) resisted the natural extension of the classical theory to the microscopicdegrees of freedom. After Planck, Einstein, Bohr, and Sommerfeld, there hadformed a simple constrained version of the classical theory. The naively quantizedclassical dynamics was already able to describe the non-continuous (discrete)spectrum of stationary states of the microscopic degrees of freedom. But thedetailed dynamics of the transitions between the stationary states was not con-tained in this theory. Nonetheless, the successes (e.g., the description of spectrallines) shaped already the dichotomous physics world concept: the microscopicdegrees of freedom obey other laws than macroscopic ones do. After theachievements of Schrödinger, Heisenberg, Born, and Jordan, the quantum theoryemerged to give the complete description of the microscopic degrees of freedom inperfect agreement with experience. This quantum theory was not a mere quantizedversion of the classical theory anymore. Rather it was a totally new formalism ofcompletely different structure than the classical theory, which was applied pro-fessedly to the microscopic degrees of freedom. As for the macroscopic degrees offreedom, one continued to insist on the classical theory.

L. Diósi, A Short Course in Quantum Information Theory,Lecture Notes in Physics, 827, DOI: 10.1007/978-3-642-16117-9_1,� Springer-Verlag Berlin Heidelberg 2011

1

For a sugar cube, the center of mass motion is a macroscopic degree of free-dom. For an atom, it is microscopic. We must apply the classical theory to thesugar cube, and the quantum theory to the atom. Yet, there is no sharp boundary ofwhere we must switch from one theory to the other. It is, furthermore, obvious thatthe center of mass motion of the sugar cube should be derivable from the center ofmass motions of its atomic constituents. Hence a specific inter-dependence existsbetween the classical and the quantum theories, which must give consistent res-olution for the above dichotomy. The von Neumann ‘‘axiomatic’’ formulation ofthe quantum theory represents, in the framework of the dichotomous physics worldconcept, a description of the microworld maintaining the perfect harmony with theclassical theory of the macroworld.

Let us digress about a natural alternative to the dichotomous concept.According to it, all macroscopic phenomena can be reduced to a multitude ofmicroscopic ones. Thus in this way the basic physical theory of the universe wouldbe the quantum theory, and the classical dynamics of macroscopic phenomenashould be deducible from it, as limiting case. But the current quantum theory is notcapable of holding its own. It refers to genuine macroscopic systems as well, thusrequiring classical physics as well. Despite the theoretical efforts in the second halfof the twentieth century, so far there has not been consensus regarding the (uni-versal) quantum theory which would in itself be valid for the whole physicalworld.

This is why we keep the present course of lectures within the framework of thedichotomous world concept. The ‘‘axiomatic’’ quantum theory of von Neumannwill be used. Among the bizarre structures and features of this theory, discreteness(quantumness) was the earliest, and the theory also drew its name from it. Yetanother odd prediction of quantum theory is the inherent randomness of themicroworld. During the decades, further surprising features have come to light. Ithas become ‘‘fashion’’ to deduce paradoxical properties of quantum theory. Thereis a particular range of paradoxical predictions (Einstein–Podolsky–Rosen, Bell)which exploits such correlations between separate quantum systems which couldnever exist classically. Another cardinal paradox is the non-clonability of quantumstates, meaning the fidelity of possible copies will be limited fundamentally andstrongly.

The initial role of the paradoxes was better knowledge of quantum theory. Welearned the differenciae specificae of the quantum systems with respect to theclassical ones. The consequences of the primarily paradoxical quantumness areunderstood relatively well and also their advantage is appreciated with respect toclassical physics (see, e.g., semiconductors, superconductivity, superfluidity). Bythe end of the twentieth century the paradoxes related to quantum-correlationshave come to the front. We started to discover their advantage only in the pastdecade. The keyword is: information! Quantum correlations, consequent uponquantum theory, would largely extend the options of classical informationmanipulation including information storage, coding, transmitting, hiding, pro-tecting, evaluating, as well as algorithms and game strategies. All these represent

2 1 Introduction

the field of quantum information theory in a wider sense. Our short course coversthe basic components only, at the introductory level.

Chapters 2–4 summarize the classical, the semiclassical, and the quantum physics.The Chaps. 2 and 4 look almost like mirror images of each other. I intended to exploitthe maximum of existing parallelism between the classical and quantum theories, andto isolate only the essential differences in the present context. Chapter 5 introduces thetext-book theory of abstract two-state quantum systems. Chapter 6 discusses theirquantum informatic manipulations and presents two applications: copy-protection ofbanknotes and of cryptographic keys. Chapter 7 is devoted to composite quantumsystems and quantum correlations (also called entanglement). An insight into threetheoretical antecedents is discussed and finally I show two quantum informaticapplications: superdense coding and teleportation. Chapter 8 introduces us to themodern theory of quantum operations. The first parts of Chaps. 9 and 10 are againmirror images of each other. The foundations of classical and quantum informationtheories, based respectively on the Shannon and von Neumann entropies, can bedisplayed in parallel terms. This holds for the classical and quantum theories of datacompression as well. There is, however, a separate section in Chap. 10 to deal with theentanglement as a resource, and with its conversions which all make sense only inquantum context. Chapter 11 offers simple introduction to the quintessence ofquantum information which is quantum algorithms. I present the concepts that lead tothe idea of the quantum computer. Two quantum algorithms will close the chapter:solution of the oracle and of the searching problems. A short section of divers Prob-lems and Exercises follows each chapter. This can, to some extent, compensate thereader for the laconic style of the main text. A few missing or short-spoken proofs andarguments find themselves as Problems and Exercises. That gives a hint how theknowledge, comprised into the economic main text, could be derived and applied.

For further reading, we suggest the monograph by Nielsen and Chuang [1]which is the basic reference work for the time being, together with by Preskill [2]and edited by Bouwmeester, Ekert and Zeilinger [3]. Certain statements ormethods, e.g. in Chaps. 10 and 11, follow [1] or [2] and can be checked theredirectly. Our bibliography continues with textbooks [4–10] on the traditionalfields, like, e.g. the classical and quantum physics, which are necessary forquantum information studies. References to two useful reviews on q-cryptography[11] and on q-computation are also included [12]. The rest of the bibliographyconsists of a very modest selection of the related original publications [13–50] andof recent related textbooks [51–55].

References

1. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. CambridgeUniversity Press, Cambridge (2000)

2. Preskill, J.: Quantum computation and information, (Caltech 1998). http://theory.caltech.edu/people/preskill/ph229/

1.1 Introduction 3

3. Bouwmeester, D., Ekert, A., Zeilinger, A. (eds): The Physics of Quantum Information:Quantum Cryptography, Quantum Teleportation, Quantum Computation. Springer, Berlin(2000)

4. Landau, L.D., Lifshitz, E.M.: Course of Theoretical Physics: Mechanics. Pergamon, Oxford(1960)

5. Gutzwiller, M.C.: Chaos in Classical and Quantum Mechanics. Springer, Berlin (1991)6. von Neumann, J.: Mathematical Foundations of Quantum Mechanics. Princeton University

Press, Princeton (1955)7. Peres, A.: Quantum Theory: Concepts and Methods. Kluwer, Dordrecht (1993)8. Kraus, K.: States, Effects, and Operations: Fundamental Notions of Quantum Theory.

Springer, Berlin (1983)9. Busch, P., Lahti, P.J., Mittelstadt, P.: The Quantum Theory of Measurement. Springer, Berlin

(1991)10. Joos, E., Zeh, H.D., Kiefer, C., Giulini, D., Kupsch, K., Stamatescu, I.O.: Decoherence and

the Appearance of a Classical World in Quantum Theory, 2nd edn. Springer, Berlin (2003)11. Gisin, N., Ribordy, G., Tittel, W., Zbinden, H.: Rev. Mod. Phys. 74, 145 (2002)12. Ekert, A., Hayden, P., Inamori, H.: Basic Concepts in Quantum Computation, (Les Houches

lectures 2000); Los Alamos e-print arXiv: quant-ph/001101313. Aharonov, Y., Albert, D.Z., Vaidman, L.: Phys. Rev. Lett. 60,1351 (2008)14. Diósi, L.: Weak measurements in quantum mechanics. In: Françoise, J.P., Naber, G.L., Tso,

S.T. (eds) Encyclopedia of Mathematical Physics, vol. 4, pp. 276–282. Elsevier, Oxford(2006)

15. Werner, R.F.: Phys. Rev. A 40, 4277 (1989)16. Wootters, W.K., Zurek, W.K.: Nature 299, 802 (1982)17. Ivanovic, I.D.: Phys. Lett. A 123, 257 (1987)18. Wiesner, S.: SIGACT News 15, 77 (1983)19. Bennett, C.H.: Phys. Rev. Lett. 68, 3121 (1992)20. Bennett, C.H., Brassard, G.: Quantum cryptography: public key distribution and coin tossing,

In: Proceedings of IEEE International Conference on Computers, Systems and SignalProcessing, IEEE Press, New York (1984)

21. Braunstein, S.L., Mann, A., Revzen, M.: Phys. Rev. Lett. 68, 3259 (1992)22. Einstein, A., Podolsky, B., Rosen, N.: Phys. Rev. 47, 777 (1935)23. Stinespring, W.F.: Proc. Am. Math. Soc. 6, 211 (1955)24. Peres, A.: Phys. Rev. Lett. 77, 1413 (1996)25. Horodecki, M., Horodecki, P., Horodecki, R.: Phys. Lett. A 223, 1 (1996)26. Bell, J.S.: Physics 1, 195 (1964)27. Clauser, J.F., Horne, M.A., Shimony, A., Holt, R.A.: Phys. Rev. Lett. 23, 880 (1969)28. Popescu, S.: Phys. Rev. Lett. 74, 2619 (1995)29. Bennett, C.H., Wiesner, S.J.: Phys. Rev. Lett. 69, 2881 (1992)30. Bennett, C.H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Phys. Rev.

Lett. 70, 1895 (1993)31. Lindblad, G.: Commun. Math. Phys. 48, 199 (1976)32. Gorini, V., Kossakowski, A., Sudarshan, E.C.G.: J. Math. Phys. 17, 821 (1976)33. Shannon, C.E.: Bell Syst. Tech. J. 27, 379–623 (1948)34. Schumacher, B.: Phys. Rev. A 51, 2738 (1995)35. Holevo, A.S.: Problems Inf. Transm. 5, 247 (1979)36. Feynman, R.P.: Int. J. Theor. Phys. 21, 467 (1982)37. Deutsch, D.: Proc. R. Soc. Lond. A 400, 97 (1985)38. Deutsch, D., Jozsa, R.: Proc. R. Soc. Lond. A 439, 533 (1992)39. Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of

the 28th Annual STOC, Association for Computer Machinery, New York (1996)40. Shor, P.W.: Algorithms for quantum computation: discrete logarithm and factoring. In: 35th

Annual Symposium on Foundations of Computer Science. IEEE Press, Los Alamitos (1994)41. Shor, P.W.: Phys. Rev. A 52, 2493 (1995)

4 1 Introduction

42. Tucci, R.R.: QC Paulinesia. http://www.ar-tiste.com/PaulinesiaVer1.pdf (2004)43. Scarani, V., Ziman, M., Štelmanovic, P., Gisin, N., Buzek, V.: Phys. Rev. Lett. 88, 097905

(2002)44. Linden, N., Popescu, S., Skrzypczyk, P.: How small can thermal machines be? Towards the

smallest possible refrigerator, arXiv:0908.2076v1 (quant-ph) (2009)45. Alicki, R.: J. Phys. 12, 103 (1979)46. Spohn, H., Lebowitz, J.L.: Adv. Chem. Phys. 38, 109 (1979)47. Geva, E., Kosloff, R.: J. Chem. Phys. 96, 3054 (1992)48. Landauer, R.: IBM J. Res. Dev. 5, 183 (1961)49. Diósi, L., Feldmann, T., Kosloff, R.: Int. J. Quant. Inf. 4, 99 (2006)50. Csiszár, I., Hiai, F., Petz, D.: J. Math. Phys. 48, 092102 (2007)51. Jaeger, G.: Quantum Information: An Overview. Springer, Berlin Heidelberg New York

(2007)52. Bruss, D., Leuchs G.: (editors): Lectures on Quantum Information. Wiley-VCH, Weinheim

(2007)53. Stolze, J., Suter, D.: Quantum Computing: A Short Course from Theory to Experiment.

Wiley-VHC, Weinheim (2008)54. Barnett. SM.: Quantum Information. Oxford University Press, Oxford (2009)55. Bennett, C., DiVincenzo, D.P., Wootters, W.K.: Quantum Information Theory. Springer,

Berlin Heidelberg New York (2009)

References 5

Chapter 2Foundations of Classical Physics

We choose the classical canonical theory of Liouville because of the best matchwith the q-theory—a genuine statistical theory. Also this is why we devote theparticular Sect. 2.4 to the measurement of the physical quantities. Hence theelements of the present chapter will most faithfully reappear in Chap. 4 onFoundations of q-physics. Let us observe the similarities and the differences!

2.1 State Space, Equation of Motion

The state space of a system with n degrees of freedom is the phase space

C ¼ fðqk; pkÞ; k ¼ 1; 2; . . .; ng � fxk; k ¼ 1; 2; . . .; ng � fxg; ð2:1Þ

where qk, pk are the canonically conjugate coordinates of each degree of freedomin turn. The pure state of an individual system is described by the phase point �x:The generic state is mixed (Fig. 2.1), described by normalized distribution function

q � qðxÞ� 0;Z

qdx ¼ 1: ð2:2Þ

The generic state is interpreted on the statistical ensemble of identical systems.The distribution function of a pure state reads

qpureðxÞ ¼ dðx� �xÞ: ð2:3Þ

Dynamical evolution of a closed system is determined by its real Hamiltonfunction H(x). The Liouville equation of motion takes this form1:

1 The form dq/dt is used to match the tradition of q-theory notations, c.f. Chap. 4, it stands foroqðx; tÞ=ot.

L. Diósi, A Short Course in Quantum Information Theory,Lecture Notes in Physics, 827, DOI: 10.1007/978-3-642-16117-9_2,� Springer-Verlag Berlin Heidelberg 2011

7

ddt

q ¼ fH; qg; ð2:4Þ

where {�, �} stands for the Poisson brackets. For pure states, this yields theHamilton equation of motion

d�x

dt¼ �fH;�xg; H ¼ Hð�xÞ: ð2:5Þ

Its solution �xðtÞ � Uð�xð0Þ; tÞ represents the time-dependent invertible map U(t) ofthe state space. It enables us to construct the solution of the Liouville equation(2.4):

qðtÞ ¼ qð0Þ � U�1ðtÞ � MðtÞqð0Þ; ð2:6Þ

where MðtÞ denotes the corresponding reversible operation. Below is introducedthe generic notion of classical operation.

2.2 Operation, Mixing, Selection

Let operationM on a given state q mean that we perform the same transformationon each system of the corresponding statistical ensemble. Mathematically, M islinear norm-preserving map of positive kernel, transforming an arbitrary state qinto a new state Mq:

Randomly mixing the elements of two ensembles of states q1 and q2 withrespective rates w1 C 0 and w2 C 0 yields the new ensemble of state:

q ¼ w1q1 þ w2q2; w1 þ w2 ¼ 1: ð2:7Þ

Fig. 2.1 Urn model of statistical ensemble. To visualize the statistical ensemble for a state q, weimagine an urn to contain a very large number of copies of the given classical system. Thepreparation of such ensemble is never unique if the state is mixed

Fig. 2.2 Urn model of operation. We draw a system of state q from the first urn, let it go throughthe transformator M and place it into the second urn. We repeat this procedure to reach a verylarge number of copies in the second urn which will then represent the transformed state Mq

8 2 Foundations of Classical Physics

A generic mixed state can always be prepared (i.e. decomposed) as the mixture oftwo or more other mixed states in infinitely many different ways. After mixing,however, it is totally impossible to distinguish in which way the mixed state wasprepared. It is crucial, of course, that mixing must be probabilistic. A given mixedstate can also be prepared (decomposed) as a mixture of pure states and thismixture is unique.

Selection of a given ensemble into specific sub-ensembles, a contrary process tomixing, will be possible via so-called selective operations. They correspondmathematically to norm-reducing positive maps. The most typical selectiveoperations are called measurements, c.f. Sect. 2.4.

2.3 Linearity of Non-Selective Operations

The operation’s categorical linearity follows from the linearity of the procedure ofmixing (2.7). Obviously we must arrive at the same state if we mix two states firstand then we subject the systems of the resulting ensemble to the operationM or,alternatively, we perform the operation prior to mixing the two ensembles together:

M w1q1 þ w2q2ð Þ ¼ w1Mq1 þ w2Mq2: ð2:8Þ

This is just the mathematical expression of the operation’s linearity.

Fig. 2.3 Urn model of mixing. We draw a system of state q1 or q2 from the two urns on the lhschosen randomly at respective probabilities w1 and w2; we place it into the urn on the rhs. Werepeat this procedure to reach a very large number of copies in the urn on the rhs, which will thenrepresent the mixed state q = w1q1 ? w2q2

Fig. 2.4 Urn model of selection. We draw a system of state q from the urns on the lhs and let itgo through the selective operation, e.g., a measurement apparatus; we place the system into thefirst or second urn on the rhs according to the outcome 1 or 2 of the measurement. We repeat thisprocedure to reach a very large number of copies in the urns on the rhs, which will then representthe selected states q1 and q2, respectively

2.2 Operation, Mixing, Selection 9

2.4 Measurements

Consider a partition {Pk} of the phase space. The functions Pk(x) are indicator-functions over the phase space, taking values 0 or 1. They form a complete set ofpairwise disjoint functions:

Xk

Pk � 1; PkPl ¼ dklPk: ð2:9Þ

We consider the indicator functions as binary physical quantities. The wholevariety of physical quantities is represented by real functions A(x) on the phasespace. Each physical quantity A possesses, in arbitrary good approximation, thestep-function expansion

AðxÞ ¼X

k

AkPkðxÞ; k 6¼ l) Ak 6¼ Al: ð2:10Þ

The real values Ak are step-heights of the function A(x), and {Pk} is a partition ofthe phase space according to them.

The projective partition (2.9) can be generalized. We define a positivedecomposition of the constant function:

1 ¼X

n

PnðxÞ; PnðxÞ� 0: ð2:11Þ

The elements of the positive decomposition, also called effects, are non-negativefunctions PnðxÞ. They need be neither disjoint functions nor indicator-functions atall. They are, in a sense, the unsharp version of indicator-functions.

2.4.1 Projective Measurement

On each system in a statistical ensemble of state q, we can measure the simul-taneous values of the indicator-functions Pk of a given partition (2.9). Theoutcomes are random. One of the binary quantities, say Pk, is 1 with probability

pk ¼Z

Pkqdx; ð2:12Þ

while the rest of them is 0:

P1 P2 . . . Pk�1 Pk Pkþ1 . . .# # # # #0 0 . . . 0 1 0 . . .

: ð2:13Þ

The state suffers projection according to Bayes theorem of conditionalprobabilities:

10 2 Foundations of Classical Physics

q! qk �1pk

Pkq: ð2:14Þ

The post-measurement state qk is also called conditional state, i.e., conditioned onthe random outcome k. As a result of the above measurement we have randomlyselected the original ensemble of state q into sub-ensembles of states qk fork = 1, 2, ....

The projective measurement is repeatable. Repeated measurements of theindicator functions Pl on qk yield always the former outcomes dkl. The aboveselection is also reversible. If we re-unite the obtained sub-ensembles, the post-measurement state becomes the following mixture of the conditional states qk:

Xk

pkqk ¼X

k

pk1pk

Pkq ¼ q: ð2:15Þ

This is, of course, identical to the original pre-measurement state.By the projective measurement of a general physical quantity A we mean the

projective measurement of the partition (2.9) generated by its step-function-expansion (2.10). The measured value of A is one of the step-heights:

A! Ak; ð2:16Þ

the probability of the particular outcome being given by (2.12). The projectivemeasurement is always repeatable. If a first measurement yielded Ak on a given statethen also the repeated measurement yields Ak. We can define the non-selectivemeasured value of A, i.e., the average of Ak taken with the distribution (2.12):

hAi �X

k

pkAk ¼Z

Aqdx: ð2:17Þ

This is also called the expectation value of A in the state q.

Fig. 2.6 Non-selective measurement. The sub-ensembles of conditional post-measurement statesqk are re-united, contributing to the ensemble of non-selective post-measurement state which is,obviously, identical to the pre-measurement state q

Fig. 2.5 Selective measurement. The ensemble of pre-measurement states q is selected into sub-ensembles of conditional post-measurement states qk according to the obtained measurementoutcomes k. The probability pk coincides with the norm of the unnormalized conditional state Pkq

2.4 Measurements 11

2.4.2 Non-Projective Measurement

Non-projective measurement generalizes the projective one Sect 2.4.1. On eachsystem in a statistical ensemble of state q, we can measure the simultaneous valuesof the effects Pn of a given positive decomposition (2.11) but we lose repeatabilityof the measurement. The outcomes are random. One of the effects, say Pn, is1 with probability

pn ¼Z

Pnqdx; ð2:18Þ

while the rest of them is 0:

P1 P2 . . . Pn�1 Pn Pnþ1 . . .# # # # #0 0 . . . 0 1 0 . . .

ð2:19Þ

The state suffers a change according to the Bayes theorem of conditionalprobabilities:

q! qn �1pn

Pnq: ð2:20Þ

Contrary to the projective measurements, the repeated non-projective measure-ments yield different outcomes in general. The effects Pn are not binary quantities.The individual measurement outcomes 0 or 1 provide unsharp information that canonly orient the outcome of subsequent measurements. Still, the selective non-projective measurements are reversible. Re-uniting the obtained sub-ensembles,i.e., averaging the post-measurement conditional states qn, yields the original pre-measurement state.

2.4.3 Weak Measurement, Time-Continuous Measurement

We can easily generalize the discrete set of effects to continuous sets. This gen-eralization has a merit: one can construct the unsharp measurement of an arbi-trarily chosen physical quantity A. One constructs the following set of effects:

P�AðxÞ ¼1ffiffiffiffiffiffiffiffiffiffi

2pr2p exp �ð

�A� AðxÞÞ2

2r2

" #; �1� �A�1: ð2:21Þ

These effects correspond to the unsharp measurement of A. The conditional post-measurement state will be q�AðxÞ ¼ p�1

�A P�AðxÞqðxÞ, c.f. Eq. 2.20. We interprete �Aas the random outcome representing the measured value of A at the standardmeasurement error r. The outcome probability (2.18) turns out to be the followingdistribution function:

12 2 Foundations of Classical Physics

p�A ¼Z

P�AðxÞqðxÞdx; ð2:22Þ

normalized obviously byR

p�Ad�A ¼ 1.The behaviour of the unsharp measurement simplifies in the weak measurement

limit r?? [1]. This limit means in practice that the accuracy r must be chosenmuch poorer than the maximum stochastic spread of the measured quantity A inthe given state q. Then the distribution (2.22) can be approximated by a Gaussiancentered at hAi:

p�A �1ffiffiffiffiffiffiffiffiffiffi

2pr2p exp �ð

�A� hAiÞ2

2r2

" #: ð2:23Þ

A useful simplification has been achieved at the price that the precision of a singleweak measurement is extremely poor. This incapacity can be compensated fully bya suitable large statistics of repeated weak measurements.

This is the case, e.g., in time-continuous measurement of a given quantity A,performed by monitoring a single system. Intuitively, we can consider measure-ments of A repeated at frequency 1/Dt and then we might take the infinite frequencylimit Dt?0 of very unsharp - weak - measurements. The error r of single weakmeasurements must be proportional to their frequency 1/Dt of repetition. The rate

g ¼ limDt!0;r!1

1Dtr2

ð2:24Þ

is called the strength of the continuous measurement. It is known that such aconstruction of time-continuous measurement does really work [2]. For com-pleteness, we include the resulting stochastic equations

ddt

q ¼ fH; qg � ffiffiffigp

wðA� hAiÞq; ð2:25Þ

�A ¼ hAi þ wffiffiffigp ; ð2:26Þ

w is the standard white-noise function: hw(t)i = 0 and hw(t)w(s)i = d(t - s).The Eq. 2.25 governs the evolution of the state q under monitoring the quantity A;the Eq. 2.26 shows that the time-dependent outcome (measurement signal) �A isalways centered at the expectation value in the current state apart from a white-noise whose intensity is inversely proportional to the measurement strength g.

2.5 Composite Systems

The phase space of the composite system, composed of the subsystems A and B, isthe Cartesian product of the phase spaces of the subsystems:

2.4 Measurements 13

CAB ¼ CA CB ¼ fðxA; xBÞg: ð2:27Þ

The state of the composite system is described by the normalized distributionfunction depending on both phase points xA and xB:

qAB ¼ qABðxA; xBÞ: ð2:28Þ

The reduced state of subsystem A is obtained by integration of the compositesystem’s state over the phase space of the subsystem B:

qA ¼Z

qABdxB �MqAB: ð2:29Þ

Our notation indicates that reduction, too, can be considered as an operation: itmaps the states of the original system into the states of the subsystem. The stateqAB of the composite system is the product of the subsystem’s states if and only ifthere is no statistical correlation between the subsystems. But generally there issome:

qAB ¼ qAqB þ cl. corr. ð2:30Þ

Nevertheless, the state of the composite system is always separable, i.e., we canprepare it as the statistical mixture of product (uncorrelated) states:

qABðxA; xBÞ ¼X

k

wkqAkðxAÞqBkðxBÞ; wk� 0;X

k

wk ¼ 1: ð2:31Þ

The equation of motion of the composite system reads

ddt

qAB ¼ fHAB; qABg: ð2:32Þ

The composite Hamilton function is the sum of the Hamilton functions of thesubsystems themselves plus the interaction Hamilton function:

HABðxA; xBÞ ¼ HAðxAÞ þ HBðxBÞ þ HABintðxA; xBÞ: ð2:33Þ

If HABint is zero then the product initial state remains product state, the dynamicsdoes not create correlation between the subsystems. Non-vanishing HABint doesusually create correlation. The motion of the whole system is reversible, of course.But that of the subsystems is not. In case of product initial state qA(0)qB(0), forinstance, the reduced dynamics of the subsystem A will represent the time-dependent irreversible2 operation MAðtÞ which we can formally write as

2 Note that here and henceforth we use the notion of irreversibility as an equivalent to non-invertibility. We discuss the entropic-informatic notion of irreversibility in Sect. 9.8.

14 2 Foundations of Classical Physics

qAðtÞ ¼Z

qAð0ÞqBð0Þ � U�1AB ðtÞdxB �MAðtÞqAð0Þ: ð2:34Þ

The reversibility of the composite state dynamics has become lost by the reduc-tion: the final reduced state qA(t) does not determine a unique initial state qA(0).

2.6 Collective System

The state (2.2) of a system is interpreted on the statistical ensemble of identicalsystems in the same state. We can form a multiple composite system from a bignumber n of such identical systems. This we call collective system; its state spaceis the n-fold Cartesian product of the elementary subsystems phase spaces:

C C . . .C � Cn: ð2:35Þ

The collective state reads

qðx1Þqðx2Þ. . .qðxnÞ � qnðx1; x2; . . .; xnÞ: ð2:36Þ

If A(x) is a physical quantity of the elementary subsystem then, in a natural way,one can introduce its arithmetic mean, over the n subsystems, as a collectivephysical quantity

Aðx1Þ þ Aðx2Þ þ . . .þ AðxnÞn

: ð2:37Þ

Collective physical quantities are not necessarily of such simple form. Theirmeasurement is the collective measurement. It can be reduced to independentmeasurements on the n subsystems.

2.7 Two-State System (Bit)

Consider a system of a single degree of freedom, possessing the followingHamilton function:

Hðq; pÞ ¼ 12

p2 þ x2

8a2q2 � a2� �2

: ð2:38Þ

The ‘‘double-well’’ potential has two symmetric minima at places q = ±a, and apotential barrier between them. If the energy of the system is smaller than thebarrier then the system is localized in one or the other well, moving there peri-odically ‘‘from wall to wall’’. If, what is more, the energy is much smaller than thebarrier height then the motion is restricted to the narrow parts around q = a orq = -a, respectively, whereas the motion ‘‘from wall to wall’’ persists always. Inthat restricted sense has the system two-states.

2.5 Composite Systems 15

One unit of information, i.e. one bit, can be stored in it. The localized motionalstate around q = -a can be associated with the value 0 of a binary digit x, while thataround q = a can be associated with the value 1. The information storage is stillperfectly reliable if we replace pure localized states and use their mixtures instead.However, the system is more protected against external perturbations if the localizedstates constituting the mixture are all much lower than the barrier height.

The original continuous phase space (2.1) of the system has thus been restrictedto the discrete set x = {0,1} of two elements. Also the states (2.2) have becomedescribed by the discrete distribution q(x) normalized as

Pxq(x) = 1. There are

only two pure states (2.3), namely dx0 or dx1. To treat classical information, theconcept of discrete state space will be essential in Chap. 9. In the general case, weuse states q(x) where x is an integer of, say, n binary digits. The correspondingsystem is a composite system of n bits.

2.8 Problems, Exercises

2.1 Mixture of pure states. Let q be a mixed state which we mix from pure states.What are the weights we must take for the pure states, respectively? Let usstart the solution with the two-state system.

2.2 Probabilistic or deterministic mixing? What happens if the mixing is notrandomly performed? Let the target state of mixing be evenly distributed:q(x) = 1/2. Let someone mix an equal number n of the pure states dx0 and dx1,respectively. Let us write down the state of this n-fold composite system. Letus compare it with the n-fold composite state corresponding to the proper, i.e.random, mixing.

2.3 Classical separability. Let us prove that a classical composite system isalways separable. Method: let the index k in (2.31) run over the phase space(2.1) of the composite system. Let us choose k ¼ ð�xA;�xBÞ.

Fig. 2.7 Classical ‘‘two-state system’’ in double-well potential. The picture visualizes the stateconcentrated in the r.h.s. well. It is a mixture of periodic ‘‘from wall to wall’’ orbits of variousenergies that are still much smaller than the barrier height x2a2/8. One can simplify this lowenergy regime into a discrete two-state system without the dynamics. The state space becomesdiscrete consisting of two points associated with x = 0 and x = 1 to store physically what will becalled a bit x

16 2 Foundations of Classical Physics

2.4 Decorrelating a single state? Does an operationM exist such that it brings anarbitrary correlated state qAB into the (uncorrelated) product state qAqB of thereduced states qA and qB? Remember, the operation M must be linear.

2.5 Decorrelating an ensemble. Give an operation M that brings 2n correlatedstates qAB into n uncorrelated states qAqB: Mq2n

AB ¼ ðqAqBÞn. Method:consider a smart permutation of the 2n copies of the subsystem A, followed bya reduction to the suitable subsystem.

2.6 Measurement and Bayes theorem. The heart of the classical theory of esti-mation is the Bayes theorem. Let us prove that the measurement scheme isequivalent to it.

2.7 Indirect measurement. Let us prove that the non-projective measurement ofarbitrarily given effects {PnðxÞg can be obtained from projective measure-ments on a suitably enlarged composite state. Method: Construct the suitablecomposite state q(x, n) to include a hypothetical detector system to count n;perform projective measurement on the detector’s n.

References

1. Aharonov, Y., Albert, D.Z., Vaidman, L.: Phys. Rev. Lett. 60,1351 (2008)2. Diósi, L.: Weak measurements in quantum mechanics. In: Françoise, J.P., Naber, G.L., Tso,

S.T. (eds) Encyclopedia of Mathematical Physics, vol. 4, pp. 276–282. Elsevier, Oxford (2006)

2.8 Problems, Exercises 17

Chapter 3Semiclassical, Semi-Q-Physics

The dynamical laws of classical physics, given in Chap. 2, can approximatively beretained for microscopic systems as well, but with restrictions of a new type.The basic goal is to impose discreteness onto the classical theory. We add dis-cretization q-conditions to the otherwise unchanged classical canonical equations.The corresponding restrictions must be graceful in the sense that they must notmodify the dynamics of macroscopic systems and they must not destroy theconsistency of the classical equations.

Let us assume that the dynamics of the microsystem is separable in thecanonical variables (qk, pk), and the motion is finite in phase space. The canonicalaction variables are defined as

Ik �1

2p

Ipkdqk; ð3:1Þ

for all degrees of freedom k = 1, 2, … The integral is understood along one periodof the finite motion in each degree of freedom. The action variables Ik are theadiabatic invariants1 of classical motion. In classical physics they can take arbi-trary values. To impose discreteness on classical dynamics, the Bohr–Sommerfeldq-condition says that each action Ik must be an integer multiple of the Planckconstant (plus �h=2 in case of oscillatory motion):

Ik �1

2p

Ipkdqk ¼ nk þ

12

� ��h: ð3:2Þ

The integer q-numbers nk will label the discrete sequence of phase space trajec-tories which are, according to this semiclassical theory, the only possible motions.The state with n1 = n2 = … = 0 is the ground state and the excited states areseparated by finite energy gaps from it.

1 See, e.g., in Chap. VII. of Landau and Lifshitz [1].

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19

Let us consider the double-well potential (2.38) with suitable parameters suchthat the lowest states be doubly degenerate, of approximate energies �hx; 2�hx; 3�hxetc., localized in either the left- or the right-side well. The parametric condition isthat the barrier be much higher than the energy gap �hx.

Let us store 1 bit of information in the two ground states, say the ground state inthe left-side well means 0 and that in the right-side means 1. These two states areseparated from all other states by a minimum energy �hx: Perturbations of energiessmaller than �hx are not able to excite the two ground states. In this sense the abovesystem is a perfect autonomous two-state system provided the energy of itsenvironment is sufficiently low. This autonomy follows from quantization and isthe property of q-systems.

The Bohr–Sommerfeld theory classifies the possible stationary states ofdynamically separable microsystems.2 It remains in debt of capturing non-stationary phenomena. The true q-theory ( Chap. 4) will come to the decision thatthe generic, non-stationary, states emerge from superposition of the stationarystates. In case of the above two-state system, the two ground states must beconsidered as the two orthonormal vectors of a two-dimensional complex vectorspace. Their normalized complex linear combinations will represent all states ofthe two-state quantum system. This q-system and its continuum number of stateswill constitute the ultimate notion of q-bit or qubit.

3.1 Problems, Exercises

3.1 Bohr quantization of the harmonic oscillator. Let us derive the Bohr–Sommerfeld q-condition for the one-dimensional harmonic oscillator of massm = 1, bounded by the potential 1

2 x2q2:

Fig. 3.1 Stationary q-states in double-well potential. The bottoms of the wells can beapproximated by quadratic potentials 1

2 x2ðq� aÞ2: Thus we obtain the energy-level structureof two separate harmonic oscillators, one in the l.h.s. well, the other in the r.h.s. well. Thisapproximation breaks down for the upper part of the wells. Perfect two-state q-systems will berealized at low energies where the degenerate ground states never get excited

2 The modern semiclassical theory is more general and powerful, cf. Gutzwiller [2].

20 3 Semiclassical: Semi-Q-Physics

3.2 The role of adiabatic invariants. Consider the motion of the harmonicoscillator that satisfies the q-conditions with a certain q-number n. Supposethat we are varying the directional force constant x2 adiabatically, i.e., muchslower than one period of oscillation. Physical intuition says that the motion ofthe system should invariably satisfy the q-condition to good approximation,even with the same q-number n. Is that true?

3.3 Classical-like or q-like motion. There is no absolute rule to distinguishbetween microscopic and macroscopic systems. It makes more sense to ask ifa given state (motion) is q-like or classical-like. In semiclassical physics, thestate is q-like if the q-condition imposes physically relevant restrictions, andthe state is classical-like if the imposed discreteness does not practicallyrestrict the continuum of classical states. Let us argue that, in this sense, smallinteger q-numbers n mean q-like states and large ones mean classical-likestates.

References

1. Landau, L.D., Lifshitz, E.M.: Course of Theoretical Physics: Mechanics. Pergamon, Oxford(1960)

2. Gutzwiller, M.C.: Chaos in Classical and Quantum Mechanics. Springer, Berlin (1991)

3.1 Problems, Exercises 21

Chapter 4Foundations of Q-Physics

We present the standard q-theory [1] while, at each element, striving for themaximum likeness to Chap. 2 on foundations of classical physics. We go slightlybeyond the traditional treatment and, e.g., we define non-projective q-measure-ments as well as the phenomenon of entanglement. Leaf through Chap. 2 again, andcompare!

4.1 State Space, Superposition, Equation of Motion

The state space of a q-system is a Hilbert space H: In the case of d-state q-systemit is the d-dimensional complex vector space

H ¼ Cd ¼ fck; k ¼ 1; 2; . . .; dg; ð4:1Þ

where the ck’s are the elements of the complex column-vector in the given basis.The pure state of a q-system is described by a complex unit vector, also called statevector. In basis-independent abstract (Dirac-) notation it reads

jwi �

c1

c2

::

cd

266664

377775; hwj � c�1; c

�2; . . .; c�d

� �;Xd

k¼1

jckj2 ¼ 1: ð4:2Þ

The inner product of two vectors is denoted by hw|ui. Matrices are denoted by a‘‘hat’’ over the symbols, and their matrix elements are written as hwjAjui. Inq-theory, the components ck of the complex vector are called probability ampli-tudes. Superposition, i.e. normalized complex linear combination of two or morevectors, yields again a possible pure state.

L. Diósi, A Short Course in Quantum Information Theory,Lecture Notes in Physics, 827, DOI: 10.1007/978-3-642-16117-9_4,� Springer-Verlag Berlin Heidelberg 2011

23

The generic state is mixed, described by trace-one positive semidefinite densitymatrix

q ¼ qkl; k; l ¼ 1; 2; . . .; d� �

� 0; tr q ¼ 1: ð4:3Þ

The generic state is interpreted on the statistical ensemble of identical systems.The density matrix of the pure state (4.2) is a special case, it is the one-dimensionalhermitian projector onto the subspace given by the state vector:

qpure ¼ P ¼ jwihwj: ð4:4Þ

We can see that multiplying the state vector by a complex phase factor yields thesame density matrix, i.e., the same q-state. Hence the phase of the state vector canbe deliberately altered, still the same pure q-state is obtained. In the conservativeq-theory, contrary to the classical theory, not even the pure state is interpreted on asingle system but on the statistical ensemble of identical systems.

Dynamical evolution1 of a closed q-system is determined by its hermitianHamilton matrix H: The von Neumann equation of motion takes this form:

dqdt¼ � i

�h½H; q�: ð4:5Þ

For pure states, this is equivalent to the Schrödinger equation of motion

djwidt¼ � i

�hHjwi: ð4:6Þ

Its solution jwðtÞi � UðtÞjwð0Þi represents a time-dependent unitary map UðtÞ ofthe Hilbert space. It enables us to construct the solution of the von Neumannequation (4.5):

qðtÞ ¼ UðtÞqð0ÞUyðtÞ � MðtÞqð0Þ; ð4:7Þ

where MðtÞ denotes the corresponding reversible q-operation. Below isintroduced the generic notion of q-operation.

Fig. 4.1 Urn model of q-statistical ensemble. To visualize the statistical ensemble for a q-stateq; we imagine an urn to contain a very large number of copies of the given q-system. Thepreparation of such ensemble is never unique if the state is mixed

1 Our lectures use the Schrödinger-picture: the q-states q evolve with t, the q-physical quantitiesA do not.

24 4 Foundations of Q-Physics

4.2 Operation, Mixing, Selection

Let operation M on a given q-state q mean that we perform the same transfor-mation on each q-system of the corresponding statistical ensemble. Mathemati-cally, M is linear trace-preserving completely positive map, cf. Sect. 8.1,transforming an arbitrary state q into a new state Mq: Contrary to classicaloperations, not all positive maps correspond to realizable q-operations, but thecompletely positive ones do Fig 4.2.

Randomly mixing the elements of two ensembles of q-states q1 and q2 atrespective rates w1 C 0 and w2 C 0 yields the new ensemble of the q-state

q ¼ w1q1 þ w2q2; w1 þ w2 ¼ 1: ð4:8Þ

A generic mixed q-state can always be prepared (i.e. decomposed) as the mixtureof two or more other mixed q-states in infinitely many different ways. Aftermixing, however, it is totally impossible to distinguish in which way the mixedq-state was prepared. It is crucial, of course, that mixing must be probabilistic.A given mixed q-state can also be prepared (decomposed) as a mixture of pureq-states and this mixture is, contrary to the classical case, not unique in general.

Selection of a given ensemble into specific sub-ensembles, a contrary process tomixing, will be possible via so-called selective q-operations. They correspondmathematically to trace-reducing completely positive maps, cf. Sect. 8.3. The mosttypical selective q-operations are called q-measurements, cf. Sect. 4.4.

Fig. 4.2 Urn model of q-operation. We draw a q-system of state q from the first urn, let itgo through the transformator M and place it into the second urn. We repeat this procedure toreach a very large number of copies in the second urn which will then represent the transformedq-state Mq

Fig. 4.3 Urn model of mixing. We draw a q-system of state q1 or q2 from the two urns on the lhschosen randomly at respective probabilities w1 and w2; we place it into the urn on the rhs. Werepeat this procedure to reach a very large number of copies in the urn on the rhs, which will thenrepresent the mixed q-state q ¼ w1q1 þ w2q2

4.2 Operation, Mixing, Selection 25

4.3 Linearity of Non-Selective Operations

The q-operation’s categorical linearity follows from the linearity of the procedureof mixing (4.8). Obviously we must arrive at the same q-state if we mix two statesfirst and then we subject the systems of the resulting ensemble to the operationMor, alternatively, we perform the operation prior to mixing the two ensemblestogether:

M w1q1 þ w2q2ð Þ ¼ w1Mq1 þ w2Mq2: ð4:9Þ

This is just the mathematical expression of the operation’s linearity.

4.4 Measurements

Consider a partition fPkg of the state space. The hermitian matrices Pk are pro-jectors on the Hilbert space, of eigenvalues 0 and 1. They form a completeorthogonal set:

Xk

Pk ¼ I; PkPl ¼ dklPk: ð4:10Þ

We consider the projectors as binary q-physical quantities. Hermitian matrices Aacting on the Hilbert space describe the whole variety of q-physical quantities.Each q-physical quantity A possesses the spectral expansion2

A ¼X

k

AkPk; k 6¼ l) Ak 6¼ Al: ð4:11Þ

Fig. 4.4 Urn model of selection. We draw a q-system of state q from the urns on the lhs and let itgo through the selective q-operation, e.g., a measurement apparatus; we place the system into thefirst or second urns on the rhs according to the outcome 1 or 2 of the measurement. We repeat thisprocedure to reach a very large number of copies in the urns on the rhs, which will then representthe selected q-states q1 and q2; respectively

2 Equivalent terminologies, like spectral or diagonal decomposition, or just diagonalization, arein widespread use.

26 4 Foundations of Q-Physics

The real values Ak are eigenvalues of the matrix A; and fPkg is a partition of thestate space according to them.

The projective partition (4.10) can be generalized. We define a positivedecomposition of the unit matrix:

I ¼X

n

Pn; Pn� 0: ð4:12Þ

The elements of the positive decomposition, also called q-effects, are non-negativematrices Pn: They need be neither orthogonal nor projectors at all. They are, in asense, the unsharp version of projectors.

4.4.1 Projective Measurement

On each q-system in a statistical ensemble of q-state q; we can measure thesimultaneous values of the orthogonal projectors Pk of a given partition (4.10). Theoutcomes are random. One of the binary quantities, say Pk; is 1 with probability

pk ¼ tr Pkq� �

; ð4:13Þ

while the rest of them is 0:

P1 P2 . . . Pk�1 Pk Pkþ1 . . .# # # # #0 0 . . . 0 1 0 . . .

: ð4:14Þ

The state suffers projection according to the projection postulate of von Neumannand Lüders:

q! qk �1pk

PkqPk: ð4:15Þ

Fig. 4.5 Selective q-measurement. The ensemble of pre-measurement q-states q is selected intosub-ensembles of conditional post-measurement q-states qk according to the obtained measure-ment outcomes k. The probability pk coincides with the trace of the unnormalized conditionalq-state PkqPk: The relative phases of the sub-ensembles qk have been irretrievably lost in amechanism called decoherence

4.4 Measurements 27

The post-measurement q-state qk is also called conditional q-state, i.e., condi-tioned on the random outcome k. As a result of the above measurement we haverandomly selected the original ensemble of q-state q into sub-ensembles of q-statesqk for k = 1, 2, ….

The projective q-measurement is repeatable. Repeated q-measurements of theorthogonal projectors Pl on qk yield always the former outcomes dkl. The aboveselection is, contrary to the classical case, not reversible. Let us re-unite theobtained sub-ensembles; the post-measurement q-state becomes the followingmixture of the conditional q-states qk :

Xk

pkqk ¼X

k

pk1pk

PkqPk ¼X

k

PkqPk; 6¼ q; ð4:16Þ

which is in general not identical to the original pre-measurement q-state. The non-selective measurement realizes an irreversible3 q-operation M:

q!X

k

PkqPk �Mq: ð4:17Þ

Q-irreversibility has its roots in the mechanism of decoherence4 which means thegeneral phenomenon when superpositions get destroyed by q-measurement or byenvironmental interaction, cf. Chap. 8.

By the projective q-measurement of a general q-physical quantity A we meanthe projective q-measurement of the partition (4.10) generated by its spectralexpansion (4.11). The measured value of A is one of the eigenvalues:

A! Ak; ð4:18Þ

the probability of the particular outcome is given by (4.13). The projectiveq-measurement is always repeatable. If a first measurement yielded Ak on a givenstate then also the repeated measurement yields Ak. We can define the

Fig. 4.6 Non-selective q-measurement. The sub-ensembles of conditional post-measurementq-states qk are re-united, contributing to the ensemble of non-selective post-measurement q-statewhich is, contrary to the classical case and because of decoherence, irreversibly different from thepre-measurement q-state q:

3 We use the notion of irreversibility as an equivalent to non-invertibility. We discuss theentropic-informatic notion of q-irreversibility in Sect. 10.8.4 Cf., e.g., the monograph by Joos et al. [2].

28 4 Foundations of Q-Physics

non-selective measured value of A; i.e., the average of Ak taken with the distri-bution (4.13):

hAi �X

k

pkAk ¼ tr Aq� �

: ð4:19Þ

This is also called the expectation value of A in the state q:

4.4.2 Non-Projective Measurement

Non-projective q-measurement generalizes the projective one 4.4.1. On eachq-system in a statistical ensemble of state q; we can measure the simultaneousvalues of the q-effects Pn of a given positive decomposition (4.12) but we loserepeatability of the measurement. The outcomes are random. One of the q-effects,say Pn; is 1 with probability

pn ¼ tr Pnq� �

; ð4:20Þ

while the rest of them is 0:

P1 P2 . . . Pn�1 Pn Pnþ1 . . .# # # # #0 0 . . . 0 1 0 . . .

ð4:21Þ

The state suffers the following change5:

q! qn �1pn

P1=2n qP1=2

n : ð4:22Þ

Contrary to the projective q-measurements, the repeated non-projectiveq-measurements yield different outcomes in general. The q-effects Pn are notbinary quantities. The individual measurement outcomes 0 or 1 provide unsharpinformation that can only orient the outcome of subsequent measurements. Theselective non-projective q-measurements are obviously irreversible: re-unitingthe obtained sub-ensembles, i.e., averaging the post-measurement conditionalq-states qn yields

Pn P1=2

n qP1=2n which differs from the original pre-measure-

ment q-state q:

5 Note that, in the q-literature, the post-measurement states are usually specified in a more

general form ð1=pnÞUnP1=2n qP1=2

n Uyn ; to include the arbitrary selective post-measurement

unitary transformations Un:

4.4 Measurements 29

4.4.3 Weak Measurement, Time-Continuous Measurement

We can easily generalize the discrete set of q-effects to continuous sets. Thisgeneralization has a merit: one can construct the unsharp measurement of anarbitrarily chosen quantity A: One constructs the following set of q-effects:

P�A ¼1ffiffiffiffiffiffiffiffiffiffi

2pr2p exp �ð

�A� AÞ2

2r2

" #; �1� �A�1: ð4:23Þ

These q-effects correspond to the unsharp q-measurement of A: The conditional

post-measurement q-state will be q�A ¼ p�1�A P1=2

�AqP1=2

�A; cf. Eq. (4.22). We inter-

prete �A as the random outcome representing the measured value of A at thestandard measurement error r. The outcome probability (4.20) turns out to be thefollowing distribution function:

p�A ¼ tr P�Aq� �

; ð4:24Þ

normalized obviously byR

p�Ad�A ¼ 1:The behaviour of the unsharp q-measurement simplifies in the weak measure-

ment limit r ? ? [3] . This limit means in practice that the accuracy r must bechosen much poorer than the maximum stochastic spread of the measured quantityA in the given q-state q: Then the distribution (4.23) can be approximated by aGaussian centered at hAi :

p�A �1ffiffiffiffiffiffiffiffiffiffi

2pr2p exp �ð

�A� hAiÞ2

2r2

" #: ð4:25Þ

A useful simplification has been achieved at the price that the precision of a singleweak measurement is extremely poor. This incapacity can be compensated fully bya suitable large statistics of repeated weak measurements.

This is the case, e.g., in time-continuous q-measurement of a given quantity A;performed by monitoring a single q-system. Intuitively, we can consider mea-surements of A repeated at frequency 1/Dt and then we might take the infinitefrequency limit Dt ? 0 of very unsharp—weak—measurements. The error r ofsingle weak measurements must be proportional to their frequency 1/Dt of repe-tition. The rate

g ¼ limDt!0;r!1

1Dtr2

ð4:26Þ

is called the strength of the continuous measurement. It is known that such aconstruction of time-continuous measurement does really work [4]. Rather thanpresenting the equations of the selective time-continuous q-measurement, this time

30 4 Foundations of Q-Physics

we consider the non-selective case which is much simpler. Under the abovemechanism of continuous measurement of A the reversible equation of motion(4.5) turns into the following irreversible q-master equation:

dqdt¼ � i

�hH; q� �

� g

8A; A; q� �� �

: ð4:27Þ

The new double-commutator term on the rhs describes the decoherence caused bythe continuous q-measurement. This is a special case of the general q-masterequations presented in Sect. 8.6.

Let us see how this new term comes about. Note first that, in the asymptotics ofthe continuous limit (4.26), the error r in the q-effects (4.23) can be expressedthrough the measurement strength g and the time-step Dt:

P�A ¼ffiffiffiffiffiffiffiffigDt

2p

rexp � g

2ðffiffiffiffiffiDtp

�A�ffiffiffiffiffiDtp

AÞ2h i

: ð4:28Þ

Each time the unsharp measurement of A happens we can write the non-selectivechange of the q-state in the following form:

q!Z

P1=2�A

qP1=2�A

d�A: ð4:29Þ

Then we substitute the q-effects (4.28) and change the integration variable from �A

to a ¼ffiffiffiffiffiDtp

�A; yielding the following expression for the post-measurement state:ffiffiffiffiffiffig

2p

r Zexp � g

4ða�

ffiffiffiffiffiDtp

AÞ2h i

q exp � g

4ða�

ffiffiffiffiffiDtp

AÞ2h i

da: ð4:30Þ

Since we are interested in the continuous limit, we can restrict the aboveexpression for the leading term q� ðg=8Þ A; A; q

� �� �Dt in the small Dt. Hence, in

the continuous limit Dt ? 0, we arrive at the new double-commutator contributionto dq=dt:

4.4.4 Compatible Physical Quantities

Let A and B be two arbitrary q-physical quantities. Consider their spectralexpansions (4.11)

A ¼X

k

AkPk; B ¼X

l

BlQl: ð4:31Þ

Let us measure both q-physical quantities, A first and then B; in subsequent pro-jective measurements. Write down the selective change of the q-state (4.16):

4.4 Measurements 31

q! PkqPk

pk! QlPkqPkQl

plk; ð4:32Þ

where plk is the probability that A! Ak is the first, then B! Bl is the secondmeasurement outcome. Obviously, had we performed the two measurements in thereversed order, then the distribution of the measurement outcomes would ingeneral be different. This we interpret in such a way that A and B are notsimultaneously measurable. Yet, in some important cases, the measurement out-comes are independent of the order of measurements. The corresponding conditionis that the matrices of the two q-physical quantities commute:

Pk; Ql� �

¼ 0() A; B� �

¼ 0: ð4:33Þ

Q-physical quantities with commuting matrices are called compatible. On com-patible q-physical quantities one can obtain simultaneous information. Two arbi-trarily chosen q-physical quantities are, however, not compatible in general.

The matrices (projectors) of binary q-physical quantities Pk; contributing to thespectral expansion (4.11) of a given q-physical quantity, will always commute bydefinition (4.10), they are compatible, hence they can be measured simultaneously.The projective measurement defined in Sect. 4.4.1 is just the simultaneous mea-surement of all Pk’s.

Certain incompatible physical quantities can be measured in non-projectivemeasurements. The q-effects of a positive decomposition (4.12) are not compatiblein general, their non-negative matrices Pn may not commute. Yet their simulta-neous non-projective measurement 4.4.2 is possible though in restricted sensecompared to projective measurements 4.4.1: we lose repeatability of themeasurement.

4.4.5 Measurement in Pure State

General rules of q-measurement (4.13–4.18) simplify significantly for a pure stateq ¼ jwihwj: The measurement outcome is still one of the eigenvalues Ak. As forthe pure state, it jumps into a conditional pure state |wki:

A! Ak; jwi ! jwki �1ffiffiffiffiffipkp Pkjwi; ð4:34Þ

while the probability distribution of the measurement outcome reads

pk ¼ hwjPkjwi: ð4:35Þ

32 4 Foundations of Q-Physics

The description of the process becomes even simpler if the matrix of the measuredphysical quantity is non-degenerate:

A ¼ Ay ¼Xd

k¼1

AkPk () Pk ¼ jukihukj: ð4:36Þ

In this case, the state vector jumps into the unique eigenvector belonging to themeasured eigenvalue:

A! Ak; jwi ! juki: ð4:37Þ

And the probability distribution of the measurement outcome can be written as thesquared modulus of the inner product between the new and the old state vectors:

pk ¼ hwj jukihukj jwi ¼ jhukjwij2: ð4:38Þ

Most often, the description of the quantum measurement happens in such a waythat we expand the pre-measurement pure state in terms of the eigenvectors of thephysical quantity to be measured:

jwi ¼X

k

ckjuki: ð4:39Þ

In this representation the measurement takes place this way:

A! Ak;

c1

ck

cd

266664

377775!

00100

266664

377775; ð4:40Þ

while the probability distribution of the measurement outcome reads

pk ¼ jckj2: ð4:41Þ

4.5 Composite Systems

The state space of the composite q-system, composed of the q-subsystems A and B,is the tensor product of the vector spaces of the q-subsystems, cf. (2.27):

HAB ¼ CdA CdB ¼ fckl; k ¼ 1; . . .; dA; l ¼ 1; . . .; dBg: ð4:42Þ

The pure state of the composite system is described by the normalized state vectorof dimension dAdB:

4.4 Measurements 33

jwABi ¼ fcklg;Xk;l

jcklj2 ¼ 1; ð4:43Þ

If the two subsystems are uncorrelated then the state vector of the compositesystem is a tensor product

jwABi ¼ jwAi jwB ¼ fcAkcBlg: ð4:44Þ

The mixed state of the composite system is described by the dAdB 9 dAdB-dimensional density matrix (4.3)

qAB ¼ fqðklÞðlmÞg: ð4:45Þ

The reduced state of subsystem A is obtained by tracing the composite q-system’sstate over the Hilbert space of subsystem B:

qA ¼ trBqAB ¼X

l

qðklÞðllÞ

( )�MqAB: ð4:46Þ

Our notation indicates that a reduction, too, can be considered as an operationM:it maps the states of the original q-system into the states of the q-subsystem. Thestate qAB of the composite q-system is the tensor product of the q-subsystem’sstates if and only if there is no statistical correlation between the subsystems. Butgenerally there is some, and then there can be q-correlation as well. Symbolically,we write

qAB ¼ qA qB þ cl. corr.þ q-corr:; ð4:47Þ

which is different from the classical correlations (2.30). The q-correlations areabsent if and only if the state of the composite q-system is separable. In otherwords, if it can be prepared as a statistical mixture of tensor product (uncorrelated)states6:

qAB ¼X

k

wkqAk qBk; wk� 0;X

k

wk ¼ 1: ð4:48Þ

Then and only then there are no q-correlations but classical ones at most. In thecontrary case, if qAB cannot be written in the above form, then the subsystemsA and B are said to be in entangled composite state. Accordingly, q-correlation andentanglement mean exactly the same thing: the lack of classical separability.

6 This definition of q-separability was introduced by Werner [5].

34 4 Foundations of Q-Physics

The equation of motion of the composite system reads

ddt

qAB ¼ �i�h½HAB; qAB�: ð4:49Þ

The composite Hamilton matrix is the sum of the Hamilton matrices of the sub-systems themselves plus the interaction Hamilton matrix:

HAB ¼ HA IB þ IA HB þHABint: ð4:50Þ

IfHABint is zero then the tensor product initial state remains tensor product state, thedynamics does not create correlation between the q-subsystems. Non-vanishingHABint does usually create correlation. The motion of the whole q-system isreversible (unitary), of course. But that of the subsystems is not. In case of tensorproduct initial state qAð0Þ qBð0Þ; for instance, the reduced q-dynamics of thesubsystem A will represent the time-dependent irreversible q-operation MAðtÞwhich we can formally write as

qAðtÞ ¼ trB UABðtÞqAð0Þ qBð0ÞUyABðtÞ

h i�MAðtÞqAð0Þ: ð4:51Þ

The reversibility of the composite state q-dynamics has become lost by thereduction: the final reduced state qAðtÞ does not determine a unique initial stateqAð0Þ (cf. Sect. 8.2 for further discussion of reduced q-dynamics).

4.6 Collective System

The state (4.3) of a q-system is interpreted on the statistical ensemble of identicalsystems in the same state. We can form a multiple composite q-system from a bignumber n of such identical q-systems. This we call collective q-system, its statespace is the n-fold tensor product of the elementary subsystem’s vector spaces:

HH H � Hn: ð4:52Þ

The collective state reads

q q q � qn; ð4:53Þ

while the state vector of a pure collective state is

jwi jwi jwi � jwin: ð4:54Þ

If A is a q-physical quantity of the elementary subsystem then, in a natural way,one can introduce its arithmetic mean, over the n subsystems, as a collectiveq-physical quantity

A Iðn�1Þ þ I A Iðn�2Þ þ þ Iðn�1Þ A

n: ð4:55Þ

4.5 Composite Systems 35

Collective q-physical quantities are not necessarily of such simple form. Theirmeasurement is the collective q-measurement. Contrary to the classical theory, notall collective q-measurements can be reduced to independent measurements on then subsystems, cf. Sects. 7.1.4 and 10.6.

4.6.1 Problems, Exercises

4.1 Decoherence-free projective measurement. There are special conditions toavoid decoherence. Let us prove that the non-selective measurement of aq-physical quantity A does not change the measured state q if and only if½A; q� ¼ 0:

4.2 Mixing the eigenstates. Let us prove that a state given by the non-degeneratedensity matrix q can be prepared by mixing the pure eigenstates of q: Whatmixing weights shall we use? How must we generalize the method if q isdegenerate?

4.3 Weak measurement of correlation. If A and B are q-physical quantities thenfA; Bg is another q-physical quantity. We can statistically determine theexpectation value 1

2 hABþ BAi; i.e., the real part of the correlation hABi if we

measure both A and B after each other, provided at least the first measurementis weak. Let us prove that h�ABki ¼ 1

2 hABþ BAi where �A is A’s weak mea-

surement outcome, Bk is B’s projective measurement outcome.4.4 Separability of pure states. Let us prove that the pure state |wABi of a com-

posite q-system is separable if and only if it takes the form jwAi jwBi:4.5 Unitary cloning? We could try to duplicate the unknown pure state |wi, cf.

Sect. 5.3, of our q-system by cloning it to replace the prepared state |w0i ofanother q-system with the same dimension of Hilbert space. Let us prove thatthe map jwi jw0i ! jwi jwi cannot be unitary. Method: let us test whe-ther the above transformation preserves the value of inner products.

References

1. von Neumann, J.: Mathematical Foundations of Quantum Mechanics. Princeton UniversityPress, Princeton (1955)

2. Joos, E., Zeh, H.D., Kiefer, C., Giulini, D., Kupsch, K., Stamatescu, I.O.: Decoherence and theAppearance of a Classical World in Quantum Theory, 2nd edn. Springer, Berlin (2003)

3. Aharonov, Y., Albert, D.Z., Vaidman, L.: Phys. Rev. Lett. 60, 1351 (2008)4. Diósi, L.: Weak measurements in quantum mechanics. In: Françoise, J.P., Naber, G.L., Tso,

S.T. (eds) Encyclopedia of Mathematical Physics, vol. 4. Elsevier, Oxford pp. 276–282 (2006)5. Werner, R.F.: Phys. Rev. A 40, 4277 (1989)

36 4 Foundations of Q-Physics

Chapter 5Two-State Q-System: QubitRepresentations

Obviously the simplest q-systems are the two-state systems. Typical realizationsare an atom with its ground state and one of its excited states, a photon with its twopolarization states, or an electronic spin with its ‘‘up’’ and ‘‘down’’ states. Thesmallest unit of q-information, i.e. the qubit, is an abstract two-state q-system. Thischapter is technical: you learn standard mathematics of a single abstract qubit.

5.1 Computational Representation

The Hilbert space of the two-state q-system is the complex 2-dimensional vectorspace (4.1). The notion of qubit is best realized in the computational basis. Weintroduce the computational basis vectors j0i and j1i:

fjxi; x ¼ 0; 1g;Xx¼0;1

jxihxj ¼ I; hx0jxi ¼ dx0x: ð5:1Þ

Also the primitive binary q-physical quantity x is defined in the computationalbasis:

x ¼Xx¼0;1

xjxihxj ¼ j1ih1j: ð5:2Þ

This is the (singular) 2 9 2 hermitian matrix of the qubit, as q-physical quantity.Its eigenvalues are 0 and 1. Often the q-state, rather than x, is called the qubit. Thegeneric pure state is a superposition of the basis vectors:

c0j0i þ c1j1i �Xx¼0;1

cxjxi; jc0j2 þ jc1j2 ¼ 1: ð5:3Þ

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37

This is why we say the qubit carries much richer information than the classical bitdoes, since the qubit can store the values 0 and 1 in parallel as well.

It is in q-logical operations, see Sects. 6.1.1, 11.8, where the computationalbasis becomes indispensable. To explore the theoretical structure of a qubit,however, the well known physical representation is more convenient, cf. nextsection.

5.2 Pauli Representation

The mathematical models of all two-state q-systems are isomorphic to each other,regarding the state space and the physical quantities. Also the equations of motionare isomorphic for all closed two-state systems. Therefore the qubit formalism andlanguage can be replaced by the terminology of any other two-state q-system.The electronic spin is the expedient choice. This is a genuine two-state q-system,its formalism is covariant for spatial rotations which guarantees conceptual andcalculational advantage.

5.2.1 State Space

The generic pure state of a two-state q-system takes the following form in a certainorthogonal basis of ‘‘up’’ and ‘‘down’’ states:

c"j"i þ c#j#i; jc"j2 þ jc#j2 ¼ 1: ð5:4Þ

An obvious parametrization takes normalization and the free choice of thecomplex phase into the account:

cosh2j"i þ eiu sin

h2j#i: ð5:5Þ

The angular parameters h;u can be identified with the standard directional anglesof a 3-dimensional real unit vector n. This way the above q-state can be param-etrized by that unit vector itself:

jni ¼ cosh2j"i þ eiu sin

h2j#i: ð5:6Þ

All pure states of a 2-dimensional q-system have thus been brought to a one-to-onecorrespondence with the surface points of the 3-dimensional unit-sphere called theBloch sphere. The two basis q-states j"i, j#i correspond to the north and southpoles, respectively, on the Bloch sphere. Diametric points of the surface willalways correspond to a pair of orthogonal q-states: h�nj ni ¼ 0. Hence j�ni forma basis. Moreover, any basis can be brought to this form up to the phases of basis

38 5 Two-State Q-System

vectors. The modulus of the inner product of any two pure states jni; jmi is equalto the cosine of the half-angle between the two respective polarization vectorsn;m:

jhmjnij ¼ cos#

2; cos# ¼ mn: ð5:7Þ

For physical reasons, we call the 3-dimensional real Bloch vector n thepolarization vector of the pure state jni. As we shall see, the states j�ni can beinterpreted as the two eigenstates of the corresponding electronic spin-componentmatrix. As a q-physical quantity, the spin-vector r=2 of the electron was intro-duced by Pauli. Without the factor 1/2, we call r the vector of polarization. ItsCartesian components are the three Pauli matrices

rx ¼0 11 0

� �; ry ¼

0 �ii 0

� �; rz ¼

1 00 �1

� �: ð5:8Þ

The generic component of the polarization along the direction n reads

rn � nxrx þ nyry þ nzrz ¼ nr: ð5:9Þ

Some basic characteristics of the Pauli matrices are the following:

ra ¼ rya; tr ra ¼ 0; ½ra; rb� ¼ 2i�abcrc; fra; rbg ¼ 2dabI: ð5:10Þ

Let us identify the former up-down basis states (5.4) by the ‘‘spin-up’’ and ‘‘spin-down’’ eigenstates belonging to the +1 and, respectively, to the -1 eigenvalues ofthe polarization component rz:

j"i ¼ 10

� �; j#i ¼ 0

1

� �: ð5:11Þ

The polarization rn along the direction n has j� ni as eigenstates:

rn j� ni ¼ � j� ni: ð5:12Þ

This directly implies the spectral expansion of rn:

rn ¼ jnihnj � j�nih�nj: ð5:13Þ

This is why we call and denote jni as the n-up state, while the vector j�niorthogonal to it we call and denote as the n-down state:

jni � j"ni; j�ni � j#ni: ð5:14Þ

When the reference direction is one of the Cartesian axes, we use the notations likej"xi, j#xi, j"yi, j#yi, while the notations of the distinguished z-axis may some-times be omitted: j"zi � j"i; j#zi � j#i. Typically

5.2 Pauli Representation 39

j"xi ¼ 1ffiffiffi2p 1

1

� �¼ j"i þ j#iffiffiffi

2p ð5:15Þ

j"yi ¼ 1ffiffiffi2p 1

i

� �¼ j"i þ ij#iffiffiffi

2p : ð5:16Þ

5.2.2 Rotational Invariance

The general form of the 2 9 2 unitary matrices is, apart from an arbitrary complexphase, the following:

UðaÞ � exp � i2

ar

� �¼ I cos

a2� i

ar

asin

a2; ð5:17Þ

where the real a is called the vector of rotation. To interpret the name, let RðaÞdenote the orthogonal 3 9 3 matrix of spatial rotation around the direction a, bythe angle a ¼ jaj. It can be shown that the influence of the above unitary trans-formation UðaÞ on the state vector is equivalent to the spatial rotation RðaÞ of thepolarization vector:

UðaÞjni ¼ jR�1ðaÞni; ð5:18Þ

UðaÞrUyðaÞ ¼ RðaÞr: ð5:19Þ

This feature of rotational covariance makes the quick proof of many structuralproperties possible in the Pauli representation.

5.2.3 Density Matrix

The density matrix (4.4) which corresponds to the pure state jni of a two-stateq-system takes this form:

jnihnj ¼ I þ nr

2: ð5:20Þ

In the distinguished case of z-up pure state we write

j"ih"j ¼ 1 00 0

� �¼ I þ rz

2: ð5:21Þ

From here, by rotation (5.18, 5.19), we obtain the general equation (5.20). This canbe generalized even further. If we allow the polarization vector n to have lengthsshorter than 1 we can describe any mixed state as well:

40 5 Two-State Q-System

q ¼ I þ sr

2; jsj � 1: ð5:22Þ

The parameter s of the state is just the expectation value of the polarization vectorr, as q-physical quantity, in the q-state q:

s ¼ hri ¼ tr rqð Þ: ð5:23Þ

5.2.4 Equation of Motion

The general Hamilton matrix of a two-state system is the following:

H ¼ � 12

�hxr; ð5:24Þ

where x is the vector of external magnetic field provided we identify the system asthe electronic spin (and its giro-magnetic factor has been ‘‘absorbed’’ into the scaleof the magnetic field). The von Neumann equation of motion (4.5) takes this form:

dqdt¼ i

2x r;q½ �: ð5:25Þ

It implies magnetic dipole precession for the expectation value (5.23) of thepolarization vector:

ds

dt¼ �x� s: ð5:26Þ

Fig. 5.1 Bloch sphere and density matrix. The set of all possible q-states of a qubit can bevisualized by the points of a three-dimensional unit sphere of polarization vectors s. Surface pointsðjsj ¼ 1Þ correspond to pure states jsi. North and south poles are conventionally identified withj"i, j#i, respectively, whereas the polar coordinates h;u of the unit vector s coincide with those inthe orthogonal expansion (5.5) of pure states. Internal points ðjsj\1Þ correspond to mixed states.The closer s is to the center of the sphere the more mixed is the corresponding state q

5.2 Pauli Representation 41

5.2.5 Physical Quantities, Measurement

All q-physical quantities in two-state systems are real linear combinations of theunit-matrix I and the three Pauli matrices r:

A ¼ Ay ¼ a0I þ ar; ð5:27Þ

where a0 is a real number, a is a real vector. Hence the commutator of two such q-physical quantities is

½A; B� ¼ 2i a� bð Þ � r; ð5:28Þ

in obvious notation. This vanishes only when a and b are parallel vectors. Then,however, the two physical quantities A and B are simply functions of each other. Itis clear that a two-state system has no non-trivial compatible physical quantitiesSect. 4.4.4. The maximum compatible set is a single polarization rn in onedirection n; or the two orthogonal projectors P�n ¼ j� nih� nj contributing to itsspectral expansion (5.13). The three q-physical quantities rn; Pn; P�n are functionsof each other. The q-measurement of any of them is in all respects equivalent withthe measurement of any other one.

Consider a given pure state (5.6)

j"z0i ¼ cosh2j"i þ eiu sin

h2j#i; ð5:29Þ

which is a spin in direction z0 of polar angle h. Suppose the selective measurementof the distinguished polarization rz (4.37, 4.38):

j"z0i ! j"i; rz ! þ1; p" ¼ cos2ðh=2Þj#i; rz ! �1; p# ¼ sin2ðh=2Þ

�: ð5:30Þ

We can say that the initial spin of polar angle h has jumped into the verticaldirection with probability cos2(h/2). Exploiting the rotational invariance, weresume that a pure state polarized in direction n will, when a certain rm isselectively measured, jump into the new direction m with transitionprobability

pðn! mÞ ¼ jhmjnij2 ¼ cos2ð#=2Þ; cos# ¼ nm; ð5:31Þ

which is the cosine square of half the angle between the two directions n and m.This probability itself turns out to be symmetric between the initial and finalstates. We shall see in Sect. 6.2.4 of the next chapter that such transitionprobabilities can be chosen as a measure of likeness—the fidelity—between twostates in general.

42 5 Two-State Q-System

5.3 The Unknown Qubit, Alice and Bob

Because of the irreversibility of q-measurements an unknown q-state represents anissue much different from that of an unknown classical state. For various purposesin the forthcoming chapters, we need the precise notion of the unknown qubit.Interestingly, there is no theoretical standard for the totally unknown mixedq-states but for the subset of the pure states. In this sense define we the unknown(random) qubit as a pure state jni whose polarization n is totally random over the4p solid angle. It should not be confused with the totally mixed state q ¼ I=2which would correspond to the average state of the random qubits.

When we notice it, the concept of unknown qubit may be used in a particularsense. We can suppose more knowledge and less ignorance. To make the defini-tions more operational, it is common to personalize the one who knows the stateand the other one who does not. For instance, we shall assume that Alice preparesand gives Bob a qubit in one of the two non-orthogonal states

j"zi or j"xi; ð5:32Þ

between which she has decided upon her tossing a coin. That is all that also Bobknows but he does not know which of the two states he has actually received.We say that Bob has an unknown qubit in this sense. If interested, he must performa q-measurement.

5.4 Relationship of Computational and Pauli Representations

The computational 5.1 and the Pauli 5.2 representations can be transformed intoeach other by the conventional mapping of the corresponding bases1:

j0i ¼ j"i ¼ 10

� �; j1i ¼ j#i ¼ 0

1

� �: ð5:33Þ

The elemental qubit physical quantity (5.2) is in simple algebraic relationship withthe polarization component rz:

x ¼ I � rz

2; rz ¼ I � 2x: ð5:34Þ

The density matrices of the computational basis states can also be expressed by thepolarization rz:

1 Perhaps physicists would prefer the other way around: j#i = j#i and j1i = j"i.

5.3 The Unknown Qubit, Alice and Bob 43

j0ih0j ¼ I þ rz

2; j1ih1j ¼ I � rz

2: ð5:35Þ

The same thing in compact form reads

jxihxj ¼ I þ ð1� 2xÞrz

2; x ¼ 0; 1: ð5:36Þ

5.5 Fock Representation

When qubits are represented by two-state systems with a certain energy gap e, itmakes sense to identify the ground and the excited states by j0i and by j1i,respectively. We introduce the matrices of emission and absorption:

a ¼ j0ih1j; ay ¼ j1ih0j: ð5:37Þ

They satisfy the anti-commutation relationship

fa; ayg ¼ I: ð5:38Þ

The matrix j1ih1j is called the occupation number, in Fock representation it reads

n ¼ aya: ð5:39Þ

The occupation number appears in the form of the Hamiltonian as well:

H ¼ �aya ¼ �n: ð5:40Þ

Powers of a (and ay) vanish: a2 ¼ 0. The matrices a and ay can replace the Paulimatrices according to the following relationships:

rx ¼ aþ ay ry ¼ �iða� ayÞ rz ¼ 1� 2n: ð5:41Þ

It is also common to denote a; ay by r�, respectively:

a ¼ rx þ iry

2¼ rþ ay ¼ rx � iry

2¼ r�: ð5:42Þ

As we see, all physical quantities will be of the form

A ¼ Ay ¼ aI þ bnþ caþ cay; ð5:43Þ

where a, b are real, c is complex.The convenience of the Fock representation manifests itself in q-thermody-

namics where, as a matter of fact, the absorption and emission of the q-energy �become the distinguished transitions of the qubit state, see Chap. 12.

44 5 Two-State Q-System

5.6 Problems, Exercises

5.1 Pure state fidelity from density matrices. Let us show that the cosine rule (5.7)for the inner product hmjni of pure states can be derived in a single stepstarting from the corresponding two density matrices. Method: let us express

jhmjnij2.5.2 Unitary rotation for j"i �! j#i: What can be the rotation vector a that rotates

the state j"i into the orthogonal state j#i? Let us make a simple choice for therotation axis! Calculate the matrix of the corresponding unitary rotation UðaÞand verify the result.

5.3 Density matrix eigenvalues and -states in terms of polarization. Let us expressthe two eigenvalues and the two eigenvectors (i.e. the spectral expansion) of adensity matrix as function of the polarization vector s.

5.4 Magnetic rotation for j"i �! j#i: Determine the (constant) magnetic field x

that rotates the electronic spin from state j#i into state j"i. How long timemust the field be switched on?

5.5 Interrelated q-bit physical quantities. Write down all pairwise relationshipsfor the matrices rn; Pn; P�n.

5.6 Mixing non-orthogonal polarizations. Suppose we mix the pure states j"zi andj"xi together at rate 1/3:2/3. Write down the polarization vector s of theresulting mixed state.

5.6 Problems, Exercises 45

Chapter 6One-Qubit Manipulations

Numerous principles and methods of q-theory as well as q-information theory canalready be demonstrated on a two-state q-system, i.e., on a single qubit. Entan-glement can certainly not, we shall present it on composite systems in Chap. 7.The present chapter teaches the elements of q-state manipulation and varioussimple instances of the q-informatic approach.

6.1 One-Qubit Operations

We present the reversible logical operations on single qubits, whose combinationscan represent all unitary one-qubit operations. Then we discuss the example ofdepolarization q-operation generated by polarization measurements. We learn thenotorious polarization reflection which is not a q-operation: by classical analogy itwere realizable but it turns out not to be so. Let us note that the combinations ofone-qubit unitary operations and projective measurements can not represent allpossible one-qubit q-operations. These become available if the qubit forms aninteracting composite system with an environmental q-system, cf. Chap. 8.

6.1.1 Logical Operations

In q-physics, coherent operations are reversible, unitary transformations. If thecoherent operation is applied to an unknown q-state then the state remains invari-ably unknown after the operation. For qubits, the unitary transformations have beenmapped onto three-dimensional rotations (5.17). Now we are going to considerthose rotations which play a distinguished role for the logical operations on qubits.The Pauli matrices (5.8) themselves, too, are unitary. They are conventionally

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47

denoted by X; Y and Z whenever qubit-operations are considered, and the corre-sponding operations are called X-, Y-, Z-operations.

Among the three (non-trivial) classical one-bit logical operations the NOT isthe only reversible one. Hence, only the NOT possesses a q-analogue, namely, theX operation. The Z-operation and the H-operation (Hadamard operation) arefurther one-qubit operations that have no classical counterparts. The reason is thatthe Z-operation inverts the relative phase between the basis states j0i and j1i,while the H-operation brings both j0i and j1i into their superpositions. Phases andsuperpositions of j0i and j1i can not be interpreted classically. The elemental one-qubit logical operations and their influence are then the following:

X ¼ 0 11 0

� �;j0ij1i

�! X ! j1i

j0i

�ð6:1Þ

Z ¼ 1 00 �1

� �;j0ij1i

�! Z ! j0i

�j1i

�ð6:2Þ

H ¼ 1ffiffiffi2p 1 1

1 �1

� �;j0ij1i

�! H !

1ffiffi2p j0i þ j1ið Þ1ffiffi2p j0i � j1ið Þ

(: ð6:3Þ

Like all coherent (unitary) operations, also the above ones are reversible. All threeone-qubit operations are identical to their inverses, respectively.

The X, Z, H operations can be combined, e.g. XZHX is a combined unitaryoperation. Whether all unitary one-qubit operations can be realized by combining afew basic ones? The above three operations are not sufficient. Extend, nonetheless,the half-turn around the z-axis (Z) by the rotation at an arbitrary angle u, called thephase-operation

TðuÞ ¼ e�iu=2 00 eiu=2

� �;j0ij1i

�! TðuÞ ! e�iu=2 j0i

eþiu=2 j1i

�ð6:4Þ

Now, consider the combined operation HTðhÞH which is a rotation of the Blochsphere around the x-axis by the angle h. Let us follow the basis state j0i under thesubsequent transformations H; TðhÞ and H:

j0i ! j0i þ j1iffiffiffi2p ! e�ih=2 j0i þ e�ih=2 j1iffiffiffi

2p ! cos

h2j0i � i sin

h2j1i: ð6:5Þ

The resulting state corresponds to a Bloch vector with polar coordinates h and -p/2.We do a further rotation by an angle p/2 plus u around the z-axis. The cumulativeinfluence of the four logical q-operations is the following:

T12pþ u

� �HTðhÞH j0i ¼ e�iðu=2Þ�iðp=4Þ cos

h2j0i þ e�iu sin

h2j1i

� �: ð6:6Þ

48 6 One-Qubit Manipulations

Apart from an irrelevant complex phase, the r.h.s. shows the generic pure state (5.6)of a qubit. Furthermore, it can be proved that by the combinations of the operationsH and TðuÞ one can perform all one-qubit unitary operations. Therefore we call theH and the TðuÞ a set of universal one-qubit operations. This becomes important forthe theory of q-computation in Chap. 11.

6.1.2 Depolarization, Re-polarization, Reflection

The q-measurement is the most important operation among the irreversible ones.Its irreversibility is related to that we have acquired hidden q-information from aq-state and we can not completely re-supply it into the state, because of deco-herence. More general irreversible operations can be obtained if we combineq-measurements and unitary operations. It is important that the mapping of thedensity matrix remains linear and information on the state, cf. Chap. 10 for details,can only be obtained at the expense of irreversibility and decoherence.

We shall see later in Sect. 6.2.5 that there is a naive measurement strategy todetermine a single unknown qubit. This naive strategy is just a single measurementof polarization rn along a random direction n. The average post-measurement stateof the qubit becomes mixed. For a general state, the post-measurement polariza-tion becomes 1/3 smaller than the original polarization:

q � I þ sr

2!Mq � I þ sr=3

2: ð6:7Þ

The random polarization measurement leads to isotropic depolarization. Thecorresponding operation is obviously linear.

Consider now the opposite case and suppose that we wish to bring an unknownpartially polarized state into a totally polarized one, keeping the direction ofpolarization invariant:

q � I þ sr

2! I þ sr=jsj

2; i.e.: s! s

jsj : ð6:8Þ

This would mean non-linear transformation for the polarization vector as well asfor the density matrix. Hence the above re-polarization of the unknown qubitwould contradict the foundations of q-theory and, as a consequence, the corre-sponding operation does not exist (Sect. 4.3).

Consider finally a map T ; also called spin inversion or reflection, which isdoubtlessly linear:

q � I þ sr

2! T q � I � sr

2; i.e.: s! �s: ð6:9Þ

This map reflects the polarization of an unknown state. On pure states this map isjni ! j�ni: Such apparently innocent operation, which would classically mean the

6.1 One-Qubit Operations 49

mere inversion of the electron’s angular motion, does not exist in q-physics, it cannot be realized. The reason is that, in general, an isolated q-system is correlated withthe other q-systems of the world by classical and q-correlations. These q-correla-tions, i.e. these entanglements, will cause that not all linear maps are real operationson the space of q-states. For such inhibited operations the above linear map rep-resents a typical instance. However, the impossibility of the operation can only beexplained later in Sect. 7.2.2 by the mechanism of entanglement which assumescomposite systems.

6.2 State Preparation, Determination

We start with standard applications and proceed to the paradoxical ‘‘no-cloning’’theorem of q-theory [1]. From this unusual property we derive two applicationswhere q-physics guarantees such security of information processing that classicalphysics can never do.

6.2.1 Preparation of Known State, Mixing

A certain demanded state, e.g., the pure state j"i polarized in the z-direction, canbe prepared from an unknown state if we simply measure the polarization rz: If theraw state was a totally random pure state then in half of the cases the measurementoutcome is +1 and the resulting state is the demanded j"i. On average, in the otherhalf of the cases the outcome is -1. Then we rotate the polarization of j#i by 180degrees via the unitary transformation rx (5.17):

j#i �! rxj#i ¼ j"i; ð6:10Þ

so that again we obtain the desired state j"i. The pure states prepared in the aboveway on demand will serve for preparation of arbitrarily prescribed mixed states.

A generic mixed state (5.22) can be expanded in this form:

q ¼ I þ sr

2¼ 1þ jsj

2j"sih"sj þ 1� jsj

2j# sih# sj: ð6:11Þ

We can thus mix the desired state q from s-up and s-down1 pure states (5.12). Butwe can do it from other two or more pure states. Even mixed states can be used asraw material for mixing. Let us find the general rule! The basic Eq. (4.8) of mixingtwo q-states yields the following form for two-state density matrices (5.22):

I þ sr ¼ w1ðI þ s1rÞ þ w2ðI þ s2rÞ: ð6:12Þ

1 The notations j"si; j#si mean j"ni; j#ni, resp., where n ¼ s=jsj.

50 6 One-Qubit Manipulations

Thus the relationship of the polarization vectors must be this:

s ¼ w1s1 þ w2s2: ð6:13Þ

The rule can be generalized for multiple weighted mixing:

q ¼X

k

wkqk () s ¼X

k

wksk: ð6:14Þ

The polarization vector of the mixture is identical to the weighted mean ofpolarization of the components. The Bloch vector of the mixture can therefore beanywhere within the convex hull of the components Bloch vectors, depending onthe distribution of the weights of mixing. Hence a pure state can never be obtainedfrom mixing any other states. True mixed states can be prepared by mixing pure oreven mixed states in infinitely many different ways.

The completely depolarized (rotationally invariant) mixed state, i.e., theensemble behind it, can be mixed from the z-up/down pure states:

q � 12

I ¼ 50% j" zi50% j# zi

�; ð6:15Þ

as well as from the x-up/down pure states:

q � 12

I ¼ 50% j"xi50% j#xi

�: ð6:16Þ

Recall Sect. 4.2 that after mixing it is totally impossible to distinguish which waythe mixed state was prepared. It is crucial, of course, that mixing always means aprobabilistic one. To prepare, e.g., the ensemble (6.15), we can use repeated cointossing to draw the z-up/down states for the mixture (Table 6.1).

6.2.2 Ensemble Determination of Unknown State

The determination of a completely unknown (not necessarily pure) qubit state q isequivalent to the determination of the polarization vector s. The expectation valueof the physical quantity s parametrizes the state q:

Table 6.1 Uniqueness of state, non-uniqueness of mixing

The mixed state is uniquely defined by its density matrix q. For a given mixed state (i.e.: for thecorresponding ensemble) there no longer exists any test to distinguish which way the mixingwas done or, which way the mixed state was altogether created.

All is the same for classical densities q. There is a cardinal departure, however. The decompositionof a classical state (ensemble) into the mixture of pure states is always unique. Differently fromthe classical mixed states, a non-pure q-state can be decomposed into various mixtures of pureq-states.

6.2 State Preparation, Determination 51

q ¼ I þ sr

2; s ¼ tr rqð Þ ¼ ? ð6:17Þ

The three components of r are the three Pauli matrices (5.8). One must measurethem on the given state q. Since they do not commute, i.e., are not compatible, onemust measure them in separate measurements. If the q-ensemble representing thestate q consists of N elements then it is possible to consume cca. N/3 elements forthe measurement of each Pauli matrix. The outcome of each measurement is either+1 or -1; up or down in other words. The obtained statistics allows for theestimation of the polarization components:

sx �N"x � N#xN"x þ N#x

; sy �N"y � N#yN"y þ N#y

; sz �N"z � N#zN"z þ N#z

; ð6:18Þ

and, from them, we can write down the estimation of q. The statistical error isconcomitant of the method, it would only disappear at infinite N.

The above method is far from being optimum. The problem is open because,first of all, there is no natural definition of ‘‘completely’’ unknown q-state. Thesituation is simpler if we know at least that the state is pure, i.e. it can be describedby a certain state vector jni. The unknown pure state has the natural definition: wesuppose it is the random complex unit vector in the Hilbert space. For two-statesystems, equivalently, we supposed in Sect. 5.3 that the polarization unit-vector nis a perfectly random unit vector on the Bloch sphere.

6.2.3 Single State Determination: No-Cloning

An unknown q-state can only be determined if we have access to a large number ofsystems in the same state. If we possess but a single system we would first trycopying it, multiplying it. If multiplication were successful, we would produce anynumber of clones and using this ensemble we would determine the state.In two-state case, we would deliberately reduce the statistical error of polarizationestimation.

In reality, however, an unknown q-state can not be cloned. In order of an indirectproof, let us consider a completely mixed q-ensemble of two-state systems. Drawan element of it at random and suppose that we can copy and multiply it inN examples. If the ensemble had been mixed from z-up/down states (6.15) then wehave N copies either of z-up or z-down states in our hands. For very large N, thismethod allows us to explore, with high reliability, which states the depolarizedensemble had been mixed from. In particular, the alternative x-up/down states canbe excluded. This would, however, contradict to the principle that a q-ensemble isuniquely and exhaustively determined by its density matrix.

In summary, the unknown q-state of a single physical system can never beimplemented on the other single system so that it, too, bear the same q-state. As aconsequence, the q-information hidden in the unknown state of a single system can

52 6 One-Qubit Manipulations

never be copied to the other single system’s q-state as well. No carbon copy,no emergency copy can be produced ever.

Suppose, for instance, Alice prepares a single q-system in a pure state. If shedoes not tell Bob of the direction of polarization then she may safely trust theq-system to Bob since he will never be able to make a copy for himself. If Bobmakes an attempt, the copy will usually be faint and Alice will notice the fraudwhen she gets back either her copy or the fake one, see Sect. 6.2.5.

6.2.4 Fidelity of Two States

When a given q-state q is approximated by another state q0 we may need tomeasure the quality of the approximation. This purpose is served by the fidelityFðq; q0Þ. If both states are pure then the definition is this:

F ¼ jhwjw0ij2 ¼ trðqq0Þ; ð6:19Þ

i.e., their fidelity is equal to the squared modulus of their inner product. Fidelityhas simple statistical interpretation for pure states. Suppose, for instance, that Bobasks Alice for a certain pure state jwi, but she passes another pure state jw0iinstead. When Bob, by the projective measurement of jwihwj, verifies whether hehas received the demanded state jwi he will find the received state faithful to thedemanded one just with the probability F.

For qubits jni; jmi, fidelity becomes equal to the squared-cosine of the half-angle between the two polarization vectors:

F ¼ jhmjnij2 ¼ cos2ðh=2Þ; cosðhÞ ¼ mn: ð6:20Þ

6.2.5 Approximate State Determination and Cloning

Let us calculate the average fidelity between an unknown qubit jni and its faintcopy jmi. Suppose Alice passes a state jni ¼ j"i to Bob. Bob needs a copy. Hedoes not know the state and, for lack of a better idea, he measures the polarizationrz0 at a randomly chosen angle z0:

j"i ! j"z0i; rz0 ! þ1; p" ¼ cos2ðh=2Þj#z0i; rz0 ! �1; p# ¼ sin2ðh=2Þ

�ð6:21Þ

and he prepares his copy jmi after the resulting state j"z0i or j#z0i according to themeasurement outcome ± 1. The average fidelity of the copy is

p"jh"j"z0ij2 þ p#jh"j#z0ij2: ð6:22Þ

6.2 State Preparation, Determination 53

Since Bob could only choose the direction z0 at random, the expected fidelity of thecopy will be the isotropic average over all directions, that is 2/3. What does it mean?If Alice gets the copy back to check it, obviously she measures rz and expects theoutcome rz ! þ1. The probability of this result is just equal to the above calcu-lated fidelity. Hence Alice confirms the copy with 66% probability whereas with33% probability she obtains the measurement outcome rz ! �1 meaning that 1/3of the copies unveils as completely wrong. Prior to Alice’s test, however, it makesno sense to say that the copy is completely true or completely flawed.

6.3 Indistinguishability of Two Non-Orthogonal States

In a particular special case of single state determination, one knows a priori thatthe unknown state is drawn from two known states. The strategy is exceptionallysimple when the two states are orthogonal pure states, e.g., we have to decidebetween

j"ni or j#ni; ð6:23Þ

where n is known. Then a single measurement of rn provides the answer. If, on thecontrary, the possible states are not orthogonal, e.g.:

j"zi or j"xi; ð6:24Þ

then there, of course, does not exist any physical quantity such that its singlemeasurement would provide the safe answer. Between two non-orthogonal states itis impossible to decide with full certainty. This is what q-banknote 6.4.1 andq-cryptography 6.4.2 will be based on. Decision of limited reliability is stillpossible. In general, the single measurement strategy can be optimized for variousaspects like, e.g., a higher fidelity 6.2.4, a higher ratio of perfect conclusion 6.3.2,or increased accessible information 10.4.

6.3.1 Distinguishing Via Projective Measurement

Let our single qubit have either of the two states j"zi; j"xi at 50–50% apriorilikelihoods. We can try a polarization measurement either in direction z or x, canmake a choice at equal rate. At 75% likelihood the outcome will be 1 and at 25%the outcome will be -1. In the former case the measurement is not conclusive.In the latter case the measurement is perfectly conclusive: if we happened tomeasure rz ¼ �1 then the pre-measurement state could not be the j"zi but j"xi.And if we measured rx ¼ �1 then the pre-measurement state must have been thej"zi. The probability of conclusive answer is 25% only. By using non-projectivemeasurement, a higher ratio of conclusive answers can be achieved [2], as it isshown in Sect. 6.3.2.

54 6 One-Qubit Manipulations

6.3.2 Distinguishing Via Non-projective Measurement

Let us introduce the following positive decomposition (4.12):

Pz ¼ wj#xih#xj; Px ¼ wj#zih#zj; P? ¼ I � Pz � Px: ð6:25Þ

If the non-projective measurement yields Pz ! 1 or Px ! 1 then the pre-mea-surement state must have been j"zi or j"xi, respectively. If the outcome is P? ! 1then no information is gained. The likelihood of such inconclusive cases followsfrom (4.20):

p? ¼ tr P?j"zih"zj þ j"xih"xj

2

� �¼ 1� w

2; ð6:26Þ

provided the two non-orthogonal states (6.24) had equal apriori probabilities.The non-projective measurement becomes optimal for the maximum weight w:

w ¼ffiffiffi2p

1þffiffiffi2p ; ð6:27Þ

i.e., when the effect P? becomes semi-definite:

Pz ¼ffiffiffi2p

1þffiffiffi2p j#xih#xj; Px ¼

ffiffiffi2p

1þffiffiffi2p j#zih#zj; P? ¼

2

1þffiffiffi2p j"nih"nj;

ð6:28Þ

where

n ¼ ð1; 0; 1Þffiffiffi2p : ð6:29Þ

Now the ratio of conclusive cases is

1� p? ¼w

2¼ 1� 1ffiffiffi

2p � 30%: ð6:30Þ

Hence the optimum non-projective measurement has increased the ratio of correctdecisions by cca. 5% with respect to projective measurements.

6.4 Applications of No-Cloning and Indistinguishability

The non-clonability 6.2.3 of an unknown q-state seems a shortcoming literally.Yet it becomes a benefit if we take a new point of view, namely, the security ofinformation. It can be verified in simple terms that at suitable circumstances thehidden q-information may offer absolute security such that is guaranteed in

6.3 Indistinguishability of Two Non-Orthogonal States 55

q-theory whereas classically it can by no means be guaranteed in principle either.The q-theory of unforgeable banknote had been the earliest example. The secureq-key distribution is, for the time being, the only q-protocol used already inpractice.

6.4.1 Q-Banknote

Suppose that Alice, on the behalf of the bank of issue, makes a prepared two-statequantum system stick to each issued banknote, each in randomly different q-statewhich is of course logged by Alice and the bank. The bank keeps the logs private.Civilians like Bob have no access to the data. Suppose Bob attempts to duplicate oreven multiplicate a banknote. Since the maximum fidelity of the copies is 2/3, cf.Sect. 6.2.5, the bank will, on average, unveil 1/3 of the forgeries while 2/3 of theforged money could be perfectly consumed. Just for this reason, as a matter ofcourse, Alice makes more than one independent qubit-marker stick to eachbanknote. N qubits reduces Bob’s chances from 2/3 to (2/3)N.

For Alice and the bank, a simple q-protocol may be the following. Eachbanknote is identified by the usual public serial number and by a private q-serial-number. The latter is this. Each banknote contains a sequence of N independenttwo-state q-systems (qubits). Each qubit is either j0i or j1i at random.2 Hence, eachbanknote carries a random sequence of alternative pure states, e.g.:

j1i; j0i; j0i; . . .; j0i; j1i; j1i: ð6:31Þ

Alice and the bank log the otherwise public serial number together with theencoded N-digit binary ‘‘q-serial-number’’. The latter as well as the logs are keptprivate. At the same time, both states j0i; j1i can be published. If they wereorthogonal to each other, Bob would make any number of perfect forges evenwithout destroying the original banknotes. The point is that Alice and the bankshould use non-orthogonal states,3 e.g. (6.24):

j0i ¼ j" zi; j1i ¼ j"xi; ð6:32Þ

then Bob can never profit from forgery. Even doing his best, Bob is not able toproduce a sufficient number of high fidelity copies. The bulk of the copied and theoriginal banknotes can not go through a test performed by Alice to verify thelogged combination of the public and q-serial numbers. The systematic proofneeds detailed statistical analysis which must extend to more sophisticated cloningstrategies which Bob might invent. Let us add that prior to Alice’s test, it makes no

2 Despite identical notations, one should not confuse j0i; j1i in this chapter with thecomputational basis in the rest of the volume.3 This idea of Wiesner from cca. 1969 remained unpublished for many years until [3].

56 6 One-Qubit Manipulations

sense to say that a single banknote is bad or good. This is hidden q-information aslong as the test is not yet done.

The task of longtime coherent preservation of arbitrary qubits is not yet solved.Therefore q-banknotes have so far not been realized since the qubit states ought tobe preserved on the banknote for an unlimited time. The no-cloning principle hasnonetheless another application which is realizable easier, as we see in Sect. 6.4.2.

6.4.2 Q-Key, Q-Cryptography

The fact of non-clonability can be utilized in secret, protected from the unau-thorized, communication of information. In this case the coherent preservation of aqubit must be guaranteed for not longer than the duration of its transmission.This duration is particularly short when qubits are transmitted by light.

The simplest classical cryptography is based on a secret-key. This may be asequence of binary digits which is known exclusively by the authorized parties(Alice and Bob) who will utilize the key to encode secret messages to each other.In course of distribution of the key between the authorized parties it may gothrough the hands of the unauthorized person (Eve) and she might unnoticedlyread and copy the key. Even if Eve does not learn the exact key she can still breakthe secrecy of the communication between Alice and Bob. Q-cryptography excelsclassical cryptography in that the unnoticed eavesdropping is impossible duringthe distribution of the secret key. The collaborative procedure of Alice and Bob inorder to establish the secret key is governed by the so-called secret-key q-protocol.Let us see the simplest two-state q-protocol.4

Alice sends a long sequence of random binary digits (raw-key) to Bob, encodedinto N non-orthogonal qubits like in case of q-banknotes (6.31, 6.32). Alice alsoallows Bob (and anybody else) to know the two non-orthogonal states j0i and j1i;like e.g. in (6.24), that she has used to encode 0 and 1 respectively. Accordingly,Bob measures each qubit in turn. For lack of a better idea, he alternates randomlybetween the measurement of the two physical quantities

rz � 2j0ih0j � I or rx � 2j1ih1j � I: ð6:33Þ

The potential outcomes of the measurements are always ±1. The averagefrequency of the -1 is 25%, and in all these cases Bob obtains exact informationabout the qubit received from Alice. When, for instance, the measurement of rz

yields -1 then the received state only can have been the j"xi, never the j"zi.Hence Bob can safely establish that Alice’s raw-key contains the 0 in the givenbinary digit. Similarly, if the outcome of the measurement of rx was -1 then Bobestablishes that the given digit of the key is 1. In such a way Bob restores 1/4 of the

4 This is Bennett’s version BB92 [4] of the original four-state BB84 protocol [5] by Bennett andBrassard. For a review on q-cryptography, cf. Gisin et al. [6].

6.4 Applications of No-Cloning and Indistinguishability 57

raw-key, this is the sift-key. Then Bob allows Alice (and anybody else) to knowwhich digits of the raw-key have contributed to the sift-key. The above protocolsupplies Alice and Bob with the same binary sequence of cca. N/4 digits, whichthey can use as the secret-key of their cryptographic communication.

In the language of communication theory, Alice sends the qubits, encoding theraw-key, through a q-channel to Bob. Further communications of the protocol, likethat describing the code qubits to Bob and the locations of the bits of the sift-key toAlice, are sent through classical channels. Both the classical and q-channels arepublic, anybody has the access to the bits or q-bits travelling through them. Yet,the resulting sift-key is protected from unnoticed eavesdropping. Eve can unno-ticedly eavesdrop the classical channel but she does not obtain any usefulknowledge regarding the sift-key. She must concentrate on the q-channel andintercept some of the qubits. If Eve eavesdrops the q-channel then Alice and Bobwill detect it because their sift-keys will not be identical. Surely, Eve can notdiscriminate two non-orthogonal states without altering the original state at thesame time. Eve can certainly bar Alice and Bob from establishing the secret key if,e.g., she eavesdrops the q-channel too aggressively so that she may as well makethe sift-keys of Alice and Bob completely uncorrelated. In this case there isnothing else left, Alice and Bob re-start the protocol.

In reality, the sift-keys of Alice and Bob may differ because of the eaves-dropping by Eve and/or because of the own transmission noise of the q-channel(we ignore the noise during the classical communications of the protocol). For-tunately, Alice and Bob can afterwards eliminate the above differences. They canapply classical algorithms: information reconciliation and privacy amplification.The former makes, actually, error correction while the latter reduces the relevancyof the information procured by the unauthorized Eve. At the end of the day, Aliceand Bob arrive at shorter keys than the original sift ones, which will, nonetheless,coincide with very high reliability while Eve’s related information will not exceeda given small value. This happens when the transmission noise, caused by theq-channel itself and/or by Eve’s attack on it, is not too high. If it is, then attemptsof information reconciliation and privacy amplification will eat up the sift keycompletely.

A simple method of information reconciliation goes like this. Using classicalpublic communication, Alice and Bob single out randomly the same two locationsk, l from their sift-keys. They calculate the parity xk � xl of the corresponding bitsand compare their results using the public classical channel. If the results aredifferent, Alice and Bob eliminates both the k0th and the l0 bits from their sift-keys.If the results are equal then they agree to drop one of the bits, say the k0th so thatEve can not acquire any further information on the key even if she listened to thepublic classical communication of Alice and Bob. The above procedure is iteratedby Alice and Bob until they see that the frequency of disagreement between theirshortened keys is already smaller than a given little threshold. If the length of thekey is still big enough, then Alice and Bob moves to privacy amplification. At theprice of further shortening the key, it reduces the information that may has earlierbeen acquired by Eve. Once again, in public classical communication, Alice and

58 6 One-Qubit Manipulations

Bob single out randomly the same two locations k, l from their keys. They cal-culate the parity xk � xl of the corresponding bits and substitute their original twobits by one new bit which is the parity. By iterating this procedure the informationof Eve, obtained earlier, will gradually decrease until Alice and Bob will with highprobability have the identical secret keys in their hands, on which Eve (or anyoneelse) can not have more information than a given (small) limit. We give some hintof the information theoretical proof in Sect. 9.7 (Table 6.2).

6.5 Problems, Exercises

1. Universality of Hadamard and phase operations. Let us show that all 2 9 2unitary matrices (upto a complex phase) can be constructed by the repeatedapplication of the Hadamard- H and the phase-operation TðuÞ. Method: useEuler angles of rigid body rotation kinematics.

2. Statistical error of qubit determination. Suppose Alice hands over N identicallyprepared qubits q to Bob but she does not tell Bob what the state q is. Let Bobestimate the polarization vector s by the simple method 6.2.2. Write down theestimation errors Dsx, Dsy, Dsz in function of s and the number of single statemeasurements. Method: determine the mean statistical fluctuation of the countsN"x;N"y;N"z.

3. Fidelity of qubit determination. Suppose Alice sends Bob a random qubit jni.Bob knows this but he does not know the state itself. Bob measures a polari-zation rm chosen along a random direction m. Let us determine the bestexpected fidelity of Bob’s state estimate.

4. Post-measurement depolarization. Alice prepares for Bob a state of polarizations which is unknown to Bob but he wishes to learn it. On the received state,Bob would measure the polarization along a random direction, for lack of abetter idea. Let us prove that after the measurement the polarization of the statereduces to s=3.

5. Anti-linearity of polarization reflection. The polarization reflection T isequivalent with time inversion and corresponds to anti-linear transformation Ton the space of state vectors. Let us verify that, upto a joint complex phasefactor, the basis vectors transform like T j"i ¼ �j#i and T j#i ¼ j"i.

Table 6.2 Two-state q-key protocol

Alice’s raw key 1 0 0 1 0 1 ...Alice’s qubits to Bob j"xi j"zi j"zi j"xi j"zi j"xi ...Bob’s measured quantities rz rx rx rz rz rx ...Bob’s measured outcomes +1 -1 +1 -1 +1 +1 ...Bob to Alice * * ...Distributed sift key 0 1 ...

6.4 Applications of No-Cloning and Indistinguishability 59

6. General qubit effects. Suppose the q-effects

Pn ¼ wn I þ anr

; n ¼ 1; 2; . . .

which form a positive decomposition for a qubit, cf. Sect. 4.4. List the nec-essary conditions on the wn’s and an’s.

References

1. Wootters, W.K., Zurek, W.K.: Nature 299, 802 (1982)2. Ivanovic, I.D.: Phys. Lett. A 123, 257 (1987)3. Wiesner, S.: SIGACT News 15, 77 (1983)4. Bennett, C.H.: Phys. Rev. Lett. 68, 3121 (1992)5. Bennett, C.H., Brassard, G.: Quantum cryptography: public key distribution and coin tossing,

In: Proceedings of IEEE International Conference on Computers, Systems and SignalProcessing, IEEE Press, New York (1984)

6. Gisin, N., Ribordy, G., Tittel, W., Zbinden, H.: Rev. Mod. Phys. 74, 145 (2002)

60 6 One-Qubit Manipulations

Chapter 7Composite Q-System, Pure State

Even the simplest q-systems cannot exhaustively be discussed without the conceptof composite systems. As we shall see, the reason is q-correlations betweenotherwise independent q-systems. While classical correlations permit separatetreatment of local systems, q-correlations will only permit this with particularlimitations. The mathematics of composite q-systems will be introduced from theaspect of q-correlations (entanglements). The reader may learn three historicalinstances of q-correlation. Two genuine q-informatic applications based onq-correlations will close the chapter.

7.1 Bipartite Composite Systems

The simplest composite system has four-states, consists of two 2-state subsys-tems, i.e., of two qubits. Peculiar features of the composite q-systems canalready be interpreted for such 2 9 2-state systems as well. Also the major partof our theoretical knowledge on q-correlations has been obtained for 2 9 2-statesystems. Yet we start with a summary of the general bipartite compositesystems, although we restrict the quantitative theory of entanglement for purecomposite states

jwABi ¼XdA

k¼1

XdB

l¼1

ckljk; Ai � jl; Bi; ð7:1Þ

where jk; Aif g and jl; Bif g are certain orthogonal bases in the respectivesubsystems A and B.

L. Diósi, A Short Course in Quantum Information Theory,Lecture Notes in Physics, 827, DOI: 10.1007/978-3-642-16117-9_7,� Springer-Verlag Berlin Heidelberg 2011

61

7.1.1 Schmidt Decomposition

The state vector jwABi of a bipartite composite q-system consists of probabilityamplitudes ckl of double indices (4.43) where the first (Greek) one labels the basis ofsubsystem A while the second (Latin) index labels the basis of subsystem B. Thereforethe state vector can formally be considered a matrix and we can apply the theorem ofSchmidt diagonalization to it: the above two orthogonal bases can be chosen in such away that the matrix ckl of amplitudes becomes diagonal, real, and non-negative.Taking normalization into account, we can write this theorem in the following form:

fcklg ¼ffiffiffiffiffiffiwkp

dkl; k; l ¼ 1; 2; . . . minfdA; dBg: ð7:2Þ

Observe that we have formally written the normalized non-negative amplitudes asthe square-roots of a normalized probability distribution wk whose interpretationbecomes clear below. The Schmidt decomposition generalizes the diagonalexpansion of hermitian matrices, cf. Sect. 4.4, for arbitrary, even non-quadraticmatrices. The rank of the given matrix coincides with the number of the nonzerodiagonal terms, called also the Schmidt number. Hence, according to the Schmidtdecomposition, the pure state of a bipartite composite q-system can always bewritten as the superposition of orthogonal tensor product state vectors

jwABi ¼X

k

ffiffiffiffiffiffiwkp jk; Ai � jk; Bi; ð7:3Þ

where the number of terms is the rank of the matrix of the composite stateamplitudes. This is at most the dimension of the ‘‘smaller’’ subsystem and in thisway it may be ‘‘much’’ less than the dimension of the composite system.

The reduced states of the two subsystems follow:

qA � trB jwABihwABjð Þ ¼X

k

wk jk; Aihk; Aj;

qB � trA jwABihwABjð Þ ¼X

l

wl jl; Bihl; Bj:ð7:4Þ

Observe that the bases and the coefficients of the Schmidt decomposition can beobtained from the eigenvectors and eigenvalues of the reduced density matricesqA; qB. Their spectra {wk} and {wl} are identical, in this sense the mixednesses ofqA and qB are identical. (Do not forget, this is only true when the state of thecomposite system is pure.) The reduced states remain pure if and only if the statevector of the composite state is of the tensor product form jwABi ¼ jwAi � jwBi,i.e., the Schmidt number is 1.

7.1.2 State Purification

An arbitrary mixed state q of a q-system can be derived by reduction from a purestate of a suitably constructed fictitious larger composite q-system. Let us form the

62 7 Composite Q-System, Pure State

composite system consisting of the q-system in question and a fictitiousenvironmental q-system E.1 Consider the spectral expansion (4.11, 4.36) of thestate q of the original q-system:

q ¼X

k

wkjkihkj: ð7:5Þ

Introducing a basis for the environmental q-system E, we form the following purestate of the composite (‘‘big’’) system:

jwbigi ¼X

k

ffiffiffiffiffiffiwkp jki � jk; Ei; ð7:6Þ

where we assume that the dimension of E is not less than the dimension of theoriginal system. By the way, the above form is a Schmidt decomposition (7.3).If we reduce it, we recover the density matrix (7.5) of the original q-system:

q ¼ trEðjwbigihwbigjÞ: ð7:7Þ

Consequently, any mixed state q can be considered as the reduction of the purestate jwbigi of an enlarged q-system. The procedure as well as the state jwbigi arecalled purification of q. An alternative axiomatic construction of q-theory cantherefore be started from pure states instead of mixed ones.

One must avoid a certain terminological inexactitude. Purification of q-statesis not a q-operation to be performed on the given q-ensemble. Rather it is amathematical construction, a nonlinear map.

7.1.3 Measure of Entanglement

Separability of composite q-systems has been defined by (4.48): the density matrixmust be a mixture of uncorrelated (tensor product) states. For pure compositestates this is only possible if the state vector is of the tensor product form

jwABi ¼ jwAi � jwBi: ð7:8Þ

Tensor product state vectors yield obviously tensor product density matrices whichare separable. But non-product state vectors are entangled. This follows from thefact that, in the separability condition (4.48), the decomposition of a pure com-posite state qAB must reduce to a single tensor product density matrix

qAB � jwABihwABj ¼ qA � qB; ð7:9Þ

and this can only be satisfied if the state vector jwABi has the product form (7.8).

1 Sometimes we call them the principal system and the ancilla, or the system and the meter incase of the indirect measurement 8.3.

7.1 Bipartite Composite Systems 63

Accordingly, the pure composite q-state is either of the tensor product formand is thus completely uncorrelated (separable) or, alternatively, it is thesuperposition of tensor products and is thus correlated (entangled). Hence, for apure composite q-state, correlations are always q-correlations, and their structuralsource is the superposition of uncorrelated (tensor product) state vectors.Therefore it makes sense to define the measure E of entanglement on theSchmidt decomposition (7.3). We postulate that the entanglement measure beequal to the Shannon entropy 9.1 of the distribution wk governing the structureof superposition:

E jwABihwABjð Þ ¼ �X

k

wk log wk; 0�E� log minfdA; dBg: ð7:10Þ

If E = 0 then all wk vanish except for a single one which is unity; the compositestate vector becomes a tensor product which is, indeed, not entangled. In thecontrary case, E takes its maximum value when the distribution wk is flat:

jwABi ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

minfdA; dBgp X

k

jk; Ai � jk; Bi; ð7:11Þ

and this state we consider maximally entangled. In this extremal case we shall seethat the composite q-system is asymptotically equivalent with E = log min{dA, dB}pairs of maximally entangled qubits in Sect. 7.1.6. The more intrinsic meaning of thepostulated relationship (7.10) between entropy and entanglement can be verifiedlater, in the possession of the rudiments of information theory in Sects. 10.5, 10.6,10.7. Still formally, we introduce the von Neumann entropy 10.1 of an arbitraryq-state:

SðqÞ ¼ �tr q log qð Þ; 0� S� log d: ð7:12Þ

The von Neumann entropy is zero for pure states and may serve as a measure ofthe mixedness of the state. But we use it here to express the measure E ofentanglement of a given composite pure state jwABi. The von Neumann entropiesof the reduced states qA; qB (7.4) of the two subsystems coincide with each otherand with the previously defined (7.10) measure of entanglement:

E jwABihwABjð Þ ¼ SðqAÞ ¼ SðqBÞ: ð7:13Þ

In summary, the entanglement of a pure bipartite composite q-state is measured bythe (coinciding) von Neumann entropies of the reduced q-states of the two sub-systems. For mixed bipartite composite q-states the criterion of separability as wellas the measure of entanglement are much more difficult and only partially known(cf. Table 7.1). There exist separable but classically correlated states; this can onlyhappen to mixed composite states, never to pure ones.

64 7 Composite Q-System, Pure State

7.1.4 Entanglement and Local Operations

How can we create entanglement? How can we bring a product state vector into asuperposition of such products? If we choose the mere dynamical way (4.49), thenwe need a nonzero interaction Hamiltonian (4.50). If there were no interaction thenthe product initial state vector would remain of the product form. The entangle-ment E (7.13) of a general initial state vector would not change either! Why, in thelack of their interaction the subsystems A and B would evolve independently ofeach other, according to their own respective Hamilton matrices HA; HB. Theirunitary evolution leaves their entropies unchanged.

By means of suitable dynamical interaction, the entanglement of A and B can ofcourse be created, modified, increased, decreased or just eliminated. However, inparticular q-informatic situations we may not assume dynamical interactionbetween the two q-subsystems because, for instance, they are far distant from eachother. Commonly, we talk about the q-subsystems A and B as the system of Aliceand Bob, respectively, who can only influence their own local systems by thegiven choice of the local Hamilton matrices HA; HB. Accordingly, we say that theHamiltonian is local if HABint ¼ 0 and is nonlocal otherwise. Usually Alice andBob are assumed to use local dynamics so they cannot influence the entanglementbetween their systems in the dynamical way.

Another elemental mean of influencing a given q-state is when we measure acertain q-physical quantity. In case of bipartite composite systems, we call aphysical quantity local if its measurement can be performed via suitable localmeasurements by Alice and Bob, respectively. Quantities A� I or I � B aretypical local ones. The following quantity is typically nonlocal:

A� Bþ A0 � B0: ð7:14Þ

By means of local measurements, Alice and Bob are still able to determine itsexpectation value but the post-measurement state will be different from what itwould be after the standard q-measurement of the nonlocal quantity itself. Itsq-measurement is not at all viable by local q-measurements.

It will be shown in Sect. 8.5 of next chapter that local operations (LO) can nevercreate or increase entanglement. Prior to the proof, we need to determine the notion ofthe most general operations 8.3. If, however, there is already some entanglement then itcan by local methods concentrated onto a given composite subsystem. The protocol,quite unusual in common q-theory, is called distillation, Sect. 10.6.

7.1.5 Entanglement of Two-Qubit Pure States

If both A and B are 2-state q-systems (qubits) in pure states and are uncorrelatedthen their composite (two-qubit) system is a pure state of the product form

7.1 Bipartite Composite Systems 65

jwABi ¼ jnAi � jnBi: ð7:15Þ

However, the general pure two-qubit state is entangled since it can be written asthe superposition of four mutually orthogonal uncorrelated states like this, e.g., inPauli representation (5.11):

c"" j"i � j"i þ c#" j#i � j"i þ c"# j"i � j#i þ c## j#i � j#i: ð7:16Þ

The same thing in computational (5.33) representation reads

Xx1¼0;1

Xx2¼0;1

cx1;x2j1i � jx2i �

X3

x¼0

cxjxi; x � x1x2: ð7:17Þ

As we see, introducing the two-digit binary number x, i.e.: 2x1 ? x2, assignsinteger labels 0, 1, 2, 3 to the four basis vectors and this allows a compoundnotation of the general pure state in 2 9 2 dimensions.

So far we have considered bases of the two-qubit system constructed as thetensor product of the respective bases of the two single-qubit systems. Non-product bases can equally be useful. In the theory of composite spins, the fol-lowing four orthogonal basis states are common:

singlet:j"#i � j#"iffiffiffi

2p ; ð7:18Þ

triplet: j""i; j"#i þ j#"iffiffiffi2p ; j##i: ð7:19Þ

We used compact notations like j"#i for j"i � j#i. The singlet and the middle oneof the triplet states are maximally entangled (7.11) while the other two tripletstates are uncorrelated. The singlet state is rotationally invariant. This can be madeexplicit if we write down its density matrix2

qðsingletÞ ¼ j"#i � j#"iffiffiffi2p h"#j � h#"jffiffiffi

2p ¼ IAB � rA � rB

4: ð7:20Þ

Accordingly, the states j"i; j#i in the expression of the singlet state vector (7.18)can be equivalently replaced by j"ni; j#ni no matter what n is.

In the Bell basis [1], all four orthogonal vectors are maximally entangled:

jU�i ¼ j""i � j##iffiffiffi2p ;

jW�i ¼ j"#i � j#"iffiffiffi2p :

ð7:21Þ

2 The factors of two r’s, when appear as r� r or rqr, etc., should be understood as spatialscalar products. E.g.: r� r ¼ rx � rx þ ry � ry þ rz � rz.

66 7 Composite Q-System, Pure State

The polarizations are completely correlated in the first two basis states andcompletely anti-correlated in the second two. The Bell basis vector jW-i coincideswith the singlet state (7.20). Another Bell basis vector takes a compound form incomputational basis (7.17):

jUþi ¼ 1ffiffiffi2p

Xx¼0;1

jxi � jxi: ð7:22Þ

Each of the four Bell states has entanglement E = 1 by definition (7.13). We usetwo-qubit systems in Bell states as standard components to build or to decomposelarge entangled bipartite systems, as we shall see in 7.1.6, 10.6, 10.7.

7.1.6 Interchangeability of Maximal Entanglements

It is desirable to show that a general bipartite system in pure state of entanglementE is equivalent with E independent copies of two-qubit systems each in Bell statesof entanglement 1. We present the proof for the special case of maximumentanglement. For the general pure entangled state, the proof 10.6, 10.7 will needfurther q-informatic notions.

Suppose a maximally entangled bipartite pure state whose Schmidt decompo-sition (7.11) reads

jwABi ¼1ffiffiffidpXd

k¼1

jk; Ai � jk; Bi; ð7:23Þ

where, for simplicity, let dA = dB = d and let E = log(d) : k be integer. Alice’sd-state system can be built up from k qubits, i.e., in the form of a k-fold compositesystem of 2-state subsystems. A straightforward correspondence between the twobases can be obtained by the k-digit binary decomposition of the label k of theoriginal basis vectors:

jk; Ai ¼ jx1; Ai � jx2; Ai � � � � � jxk; Ai; k ¼ x1x2; . . .; xk � x: ð7:24Þ

Similarly, we introduce another basis in the state space of Bob. In the new bases,the state (7.23) can be written into this form:

1ffiffiffidpXd

x¼1

jx1; Ai � jx2; Ai � � � � � jxk; Ai½ � � jx1; Bi � jx2; Bi � � � � � jxk; Bi½ �

¼ 1ffiffiffidp

Xx¼0;1

jx; Ai � jx; Bi !�k

¼ jUþi�k:

ð7:25Þ

7.1 Bipartite Composite Systems 67

In such a way, we have proved that the pure state (7.23) in d 9 d-dimension, when itpossesses maximum entanglement E = log(d), is in a suitable basis equivalent withE copies of 2 9 2-dimensional (Bell) states of maximum entanglements E = 1:

jwABi ¼ jUþi�E: ð7:26Þ

7.2 Q-Correlations History

The appreciation of unusual correlations in composite q-systems as well as theirtheoretical characterization took a lengthy time historically. In each of the forth-coming three sections, we recall a decisive theoretical discovery.

7.2.1 EPR, Einstein Nonlocality 1935

Suppose that Alice and Bob possess one qubit each. As a result of their earlierinteraction, these qubits are being in a maximally entangled state, e.g., in thesinglet state of the Bell basis (7.21):

jwABi � jW�i ¼j"#i � j#"iffiffiffi

2p ; ð7:27Þ

where, as usual, the first polarization refers to Alice’s and the second refers toBob’s qubit, respectively. Alice and Bob, together with their qubit, live far fromeach other. Imagine that Alice measures rz on her qubit. This is the random resultof the measurement:

jwABi !j"z; #zi ; rAz ! 1; pþ ¼ 1=2j#z; "zi ; rAz ! �1; p� ¼ 1=2:

�ð7:28Þ

It is clear that the state of Bob’s qubit becomes j"i or j#i with 50–50% probabilities,perfectly anti-correlated with Alice’s post-measurement state. Alternatively, if Alicemeasures rx instead of rz then Bob’s post-measurement state becomes the mixtureof j"xi and j#xi instead of j"i and j#i. At a superficial glance, the post-measurementstate of Bob’s qubit is completely different if Alice measures rx instead of rz: In sucha way Alice could transmit information to Bob via action-at-a-distance without anyphysical interaction, merely exploiting the entanglement of their qubits. We knowthat this cannot be so: Bob has access but his own qubit and its reduced density matrixqB is totally independent from whatever measurement made by Alice. In our case qB

is the maximally mixed state. Its decomposition is never unique. It is really ambig-uous, exactly like in (6.15) and (6.16), depending on Alice remote choice rz or rx tomeasure. Yet Bob can never detect the difference.

There is, however, a crucial lesson. Einstein gave a particular strong formu-lation to the notion of locality. According to the principle of Einstein locality: In a

68 7 Composite Q-System, Pure State

theory completely describing the physical reality, the action on system A cannotinfluence the description of system B if A and B are spatially separated. As we seefrom the above EPR paradox [2], the q-theory violates Einstein locality. EPRchose the following resolution: the q-theory does not give complete description ofthe physical reality. Let us discuss the underlying motivations of EPR.

If Alice and Bob shared a correlated classical composite system, the localselective measurement (2.16) by Alice would, similarly to the above q-case,influence the state (e.g.: phase–space distribution) of Bob’s remote system. This is,however, merely the consequence of the incompleteness of the description: certainparameters of the composite system have remained hidden and treated statistically.In classical physics it is simple to see that the complete description is possiblewhen the system is in pure state, i.e. all canonical variables are exactly specified.Then the description of the classical system becomes deterministic and cannotviolate Einstein locality anymore. In the q-theory, however, even pure states canviolate Einstein locality. This is why EPR consider q-theory incomplete. Whetherq-theory, too, can be made complete if we discover its hidden parameters and wespecify their values? Whether this complete theory will satisfy Einstein locality?The negative answer comes soon in Sect. 7.2.3.

7.2.2 A Non-Existing Linear Operation 1955

Consider the maximally entangled singlet state jW-i of two qubits, whose densitymatrix reads

I � I � r� r

4: ð7:29Þ

Like in case of EPR, the first qubit belongs to Alice, the second to Bob, who are farfrom each other. Define the polarization reflection (6.9) of a generic qubit, whichcould be a hypothetical linear q-operation T :

T I þ sr

2¼ I � sr

2: ð7:30Þ

Assume that Alice is able to perform such an operation and she does it as well onher qubit entangled in the singlet state. It is straightforward to see the result.The reflection changes the sign of polarization of Alice’s own Pauli matrices whileit preserves Bob’s ones:

ðT � IÞ I � I � r� r

4¼ I � I þ r� r

4: ð7:31Þ

Surprisingly, this 4 9 4 Hermitian matrix has become indefinite! For instance, itsquadratic form with the singlet state is negative:

7.2 Q-Correlations History 69

hW�j I � I þ r� r

4jW�i ¼ tr

I � I � r� r

4I � I þ r� r

4

� �¼ � 1

2: ð7:32Þ

Hence the local reflection (7.30) of the singlet state is no more a density matrix.3

It can be shown that this anomaly arises not just for the singlet state but for allentangled two-qubit states. Therefore the local (also called partial) reflectionmakes an elegant mathematical test of entanglement, this is the Peres–Horodeckitest [4, 5] (Table 7.1).

The polarization reflection T cannot be performed in reality. To interpret thereflection (7.30) of a given local qubit, say of Alice, one should assume that it isnot entangled with whatever other qubits in the Universe. Such an assumption,however, cannot be justified. On the contrary, there are classical and q-correlationsbetween local and various remote q-systems of the Universe. But while theexistence of external classical correlations does not influence the structuralproperties (the theory) of local systems, the existence of external q-correlations(entanglements) imposes serious conditions for (the theory of) local q-systems.

7.2.3 Bell Nonlocality 1964

Can we make the q-theory complete if we introduce certain hidden parameters sothat Einstein nonlocality 7.2.1 disappear from it? Bell pointed out that, like theclassical theory, also the q-theory can be made complete and deterministic viaintroducing suitable hidden parameters. (From this aspect it is irrelevant that sucha theory would be more complicated and less elegant than the standard q-theory.)Bell has, however, also proved that the obtained complete description cannotsatisfy the locality principle of Einstein. In Bell’s formulation: no local hiddenparameter model can replace the q-theory [6].

Table 7.1 The Peres–Horodecki test of separability

Let T be the reflection of a single qubit:

T Iþsr2 ¼ I�sr

2 :

Let q be a two-qubit state. If its partial reflection is non-negative:

ðT � IÞq 0;then the state q is separable. If its partial reflection is indefinite:ðT � IÞql0;then the state q is entangled.

3 This happens just because T belongs to non-completely positive maps, cf. Chap. 8, dis-covered by Stinespring [3].

70 7 Composite Q-System, Pure State

Like in case of EPR nonlocality, we suppose Alice and Bob share an ensembleof singlet states jWABi = jW-i (7.27). Let us consider two couples of localphysical quantities on the side of Bob and Alice, respectively:

A ¼ arA; A0 ¼ a0rA; B ¼ brB; B0 ¼ b0rB; ð7:33Þ

where a, a0, b, b0 are polarization unit vectors. Alice and Bob measure four dif-ferent nonlocal combinations

A� B; A0 � B; A� B0; A0 � B0: ð7:34Þ

The measurement of the expectation values of such nonlocal quantities ispossible with local measurements. For instance, the measurement of hA� Bi isequivalent with the simultaneous local measurements of A and B by Alice andBob, respectively. But the four combinations are not compatible therefore theycannot be simultaneously measured. Alice and Bob will take four independentmeasurement statistics on four randomly chosen sub-ensembles of theq-ensemble of the singlet states. Alice and Bob measure in coincidence. Theoutcome of each local measurement is ±1. Having the four statistics been taken,Alice and Bob can calculate the expectation value of the following nonlocalquantity, cf. also (7.14):

C ¼ A� Bþ A0 � Bþ A� B0 � A0 � B0: ð7:35Þ

Let us now suppose that there exist hidden parameters which completelydetermine the outcomes of all measurements. We also adopt a further, delicateand much discussed, assumption: let the hidden parameters determine the out-comes of the chosen measurements as well as what the outcomes would havebeen, if we had chosen to do one of the other q-measurements incompatible withthe one that was actually done. Let us then index the configurations of thehidden parameters by r. Each q-system of the singlet-state q-ensemble has itsr. The hidden parameters assign definite outcome ±1 to each local polarizationmeasurement (7.33):

A ¼ Ar ¼ �1; A0 ¼ A0r ¼ �1; B ¼ Br ¼ �1; B0 ¼ B0r ¼ �1: ð7:36Þ

Note that such a hidden parameter description of the measured reality wouldtrivially satisfy the Einstein-locality: any manipulation by Alice on her qubit ofhidden parameter r would not influence the description, i.e. the values Br, Br

0, of

Bob’s qubit. Hence we call the r’s local hidden parameters.The relative frequencies of the ±1 outcomes as well as their correlations

should, of course, satisfy the predictions of q-theory. For instance:

hA� Bi ¼ limN!1

1N

Xr2X

ArBr; N ¼ jXj; ð7:37Þ

7.2 Q-Correlations History 71

and similarly for the other three combinations (7.34) as well. This implies thefollowing identity:

hCi ¼ limN!1

1N

Xr

ArBr þ A0rBr þ ArB0r � A0rB

0r: ð7:38Þ

Consider the important relationship

ArBr þ A0rBr þ ArB0r � A0rB

0r ¼ �2; ð7:39Þ

which is a consequence of (7.36). Therefore the existence of the local hiddenparameter expression (7.38) imposes the following restriction on the correlation

quantity C (7.35):

�2�hCi� 2: ð7:40Þ

This is the Bell inequality.4 Note that this inequality itself is purely classical, itsderivation has nothing to do with q-theory. We can ask whether it is satisfied by

the value hCi predicted by the q-theory.

We substitute the combination (7.35) of the polarizations (7.33) in place of hCi.In singlet state we obtain

�2� � ab� a0b� ab0 þ a0b0 � 2: ð7:41Þ

This geometrical condition must be satisfied for all possible choices of the fourdirections. But it will not be. Suppose, for instance, the following four co-planardirections:

a ¼!; a0 ¼"; b ¼.; b0 ¼- : ð7:42Þ

Insert the corresponding four scalar products into the Bell inequality (7.41):

� ab� a0b� ab0 þ a0b0

¼ �cosð3p=4Þ � cosð3p=4Þ � cosð3p=4Þ þ cosðp=4Þ ¼ 2ffiffiffi2p

:ð7:43Þ

This value violates the Bell inequality. It means that in singlet state the values(7.36) of the four physical quantities cannot be chosen as a function of thehidden parameters r in such a way that the statistical predictions of theq-theory be satisfied. This is called Bell nonlocality. Separable states are alwaysBell local. But we do not know whether or not all entangled states show Bellnonlocality [8].

4 It is the version by Clauser et al. [7].

72 7 Composite Q-System, Pure State

7.3 Applications of Q-Correlations

We learn two direct applications of q-correlations: superdense coding, and tele-portation. Options of information manipulations become available which could notbe possible in classical physics.

7.3.1 Superdense Coding

In the basic setup of communication, Alice sends information through a channel toBob via classical binary data. What can they profit if Alice sends qubits, instead ofclassical bits, through a q-channel? When Alice and Bob agree that Alice encodesthe binary information into orthogonal qubits, e.g. into j"i; j#i, then Bob measuresrz and their q-communication becomes equivalent with the classical communi-cation. But q-communication can be more powerful, at least in solving certainparticular tasks.

Suppose then a q-channel from Alice to Bob which, for the time being, they canuse without limitation as well as they can communicate through a classical channelwithout limitation. They know that one year later Alice obtains classical infor-mation from somewhere and she must forward it to Bob. They also know that oneyear later the classical channel is already not available and they can only sendqubits through the q-channel. Intuition says that they have to consume one qubit toencode one classical bit. We shall see, however, that Alice and Bob can, well intime, agree upon the protocol of superdense coding which encodes two bits intoone qubit [9].

Alice prepares pairs of qubits entangled in singlet states and sends one qubit ofeach pair to Bob through the q-channel. Both Alice and Bob preserve their ownqubits. Of course, Alice neither could nor wished to send any information to Bobvia the entangled qubits. One year later, Alice learns the first two bits that she willforward to Bob via one qubit. She encodes the two bits into the four differentunitary 1-qubit operations I; X; Y ; Z. She performs the actual operation on the firstof her qubits kept with her. Then the whole composite singlet state becomes one ofthe four orthogonal Bell states (7.21), according to the scheme

ðI � IÞ jW�i ¼ jW�iðX � IÞ jW�i ¼ jU�iðY � IÞ jW�i ¼ jUþiðZ � IÞ jW�i ¼ jWþi:

ð7:44Þ

After the unitary operation, Alice sends her qubit through the q-channel to Bobwho has thus one of the four orthogonal Bell states in his hands, encoding the twobits of information according to the protocol. The information can be decoded if

7.3 Applications of Q-Correlations 73

Bob performs a projective measurement of the Bell basis. In such a way, Bobacquires the original two qubits. If Alice and Bob repeatedly perform the aboveprotocol, Alice will be able to send 2n bit information via n qubits while they haveto consume n shared singlet states.

Superdense coding does not mean that we squeeze 2 bits of information into 1qubit. In fact it means that we can encode 2 bits of information into 2 qubits insuch a way that physically we do not touch but one of the qubits. Therefore theother qubit can be transmitted to the other party in advance, say, when the classicalinformation is not yet even known. In this sense can the coding into the retainedsingle qubit be considered denser than what would be available classically. Forsuperdense coding, entanglement has obviously been instrumental.

7.3.2 Teleportation

Transmission of an unknown qubit requires a q-channel. Yet qubit transmission ispossible through classical channel if the parties share a reservoir of previouslyentangled qubits. Suppose a q-channel from Alice to Bob which, for the timebeing, they can use without limitation as well as they can communicate through aclassical channel without limitation. They know that one year later Alice receivesunknown qubits from somewhere and she must forward them to Bob. They alsoknow that one year later the q-channel is already not available and they can onlysend bits through the classical channel. Intuition says that classical communicationis not likely to transmit q-information. We shall see, however, that Alice and Bobcan, well in time, agree upon the protocol of teleportation [10] which transmits theunknown qubit from Alice to Bob. More precisely, the original qubit remains withAlice all the time while its state becomes perfectly inherited by a raw qubit of Bob.The protocol destroys the state of the original qubit, otherwise teleportation couldmake perfect cloning, too, which would be a contradiction, cf. Sect. 6.2.3.

Alice prepares pairs of qubits entangled in singlet states and sends one qubit ofeach pair to Bob through the q-channel. Both Alice and Bob preserves their ownqubits. Of course, Alice neither could nor wished to send any information to Bobvia the entangled qubits. One year later, Alice receives the first qubit

jwi ¼ a j"i þ b j#i ð7:45Þ

that she will forward to Bob via two classical bits. Consider the 3-qubit compositestate consisting of the received qubit and a singlet state shared by Alice and Bob:

jwi � jW�i ¼ a j"i þ b j#i½ � � j"#i � j#"iffiffiffi2p : ð7:46Þ

This can be re-written if we introduce the Bell basis (7.21) for the two qubits onAlice’s side:

74 7 Composite Q-System, Pure State

� 12jW�i � a j"i þ b j#i½ � þ 1

2jU�i � a j#i þ b j"i½ �

þ 12jUþi � a j#i � b j"i½ � � 1

2jWþi � a j"i � b j#i½ �:

ð7:47Þ

We see that the singlet state jW-i on Alice’s side is multiplied by the statea j"iþb j#i ¼ jwi on Bob’s side, which is just the state (7.45) received originallyby Alice to teleport. In fact all four Bell states on Alice’s side are correlated withjwi upto the trivial unitary operations I; X; Y ; Z in turn. Then Alice performs theprojective measurement of the Bell basis. There are four possible outcomesjW�i; jU�i Alice transmits 2 bits through the classical channel to inform Bob ofthe outcome. So Bob learns which transform he must apply to his qubit:

jW�i : I½a j"i þ b j#i�jU�i : X½a j#i þ b j"i�jUþi : Y ½a j#i � b j"i�jWþi : Z½a j"i � b j#i�:

ð7:48Þ

The resulting state of Bob’s qubit is precisely jwi (7.45) which was the state toteleport. If Alice and Bob repeatedly perform the above protocol, Alice will beable to send n qubits via 2n bits while they have to consume n shared singlet states.

The four possible measurement outcomes, obtained by Alice, are always totallyrandom as it follows from the coefficients of the orthogonal decomposition (7.47).Alice does not acquire any information on the qubit that she teleports. Yet sheentangles this qubit with the qubit of the shared singlet state, and this entanglementmakes then possible that, via 2 bits of classical information, the unknown qubit re-appears on Bob’s side while it becomes completely smashed on Alice’s side. Thehidden information on the teleported qubit has been travelled through the chain ofentanglements of the 3 qubits involved in the protocol.

The above teleportation protocol applies invariably to mixed states, too. Morethan that, it teleports all external q-correlations of the given qubit. Suppose that thestate to teleport has been entangled with the state of a certain environmentalq-system:

a j1; Ei � j"i þ b j2; Ei � j#i: ð7:49Þ

Once the teleportation protocol from Alice to Bob has been done, the environ-mental state becomes entangled with Bob’s state exactly the same way. The proofis algebraically identical with the previous proof of the teleportation of theunentangled state (7.45), if we make the formal substitutions

a! a j1; Ei �; b! b j2; Ei � : ð7:50Þ

Hence Bob’s qubit becomes in all respects perfectly identical with the qubit thatAlice received to teleport, including all its external q-correlations. The qubitreceived by Alice is left behind in the completely mixed state.

7.3 Applications of Q-Correlations 75

7.4 Problems, Exercises

7.1 Schmidt orthogonalization theorem. Let us prove the Schmidt decompositiontheorem (7.2) for a complex rectangular matrix c; starting from the well-known spectral expansion theorem of Hermitian matrices. Method: apply the

latter to both matrices cyc and ccy:7.2 Swap operation. Consider two identical q-systems in arbitrary uncorrelated pure

state jwi � j/i: The swap matrix is defined by Sðjwi � j/iÞ ¼ j/i � jwi: S isunitary and Hermitian. Let us prove that for two qubits

S ¼ I � I þ r� r

2:

Method: introduce the basis j""i; j##i; j"#i; j#"i.7.3 Singlet density matrix. Let us write down the density matrix of the two-qubit

singlet state in Pauli representation. Method: exploit rotational invariance orexpress qðsingletÞ through the swap S:

7.4 Local measurement of expectation values. Let us show that the expectationvalue of the nonlocal physical quantity A� Bþ A0 � B0 can be measuredlocally as well. Method: verify that the expectation values of A� B and ofA0 � B0 are locally measurable.

7.5 Local measurement of certain nonlocal quantities. A tensor product q-physicalquantity A� B may not be measured locally although its expectation value isalways measurable locally. Let us explain why the nonlocal measurement ofrz � rz is not equivalent with the local simultaneous measurement of rz � I

and I � rz: Let us prove that for the local measurability of A� B it is sufficientif the spectrum of A� B is non-degenerate.

7.6 Nonlocal hidden parameters. Let us show that the q-theoretic correlations canafter all be reproduced by nonlocal hidden parameters. Method: we completethe hidden parameter r by a switch m = 1, 2, 3, 4 marking which of thefour measuring setups is active. Then we can already assign the valuesArm, Brm, Arm

0, Brm

0suitably to each pair of qubits in the ensemble.

7.7 Does teleportation clone the qubit? Teleportation would be a perfect clonerhad the procedure not destroyed the state of the original qubit. Let us showthat the original qubit is always left behind in the totally mixed state q ¼ I=2:

References

1. Braunstein, S.L., Mann, A., Revzen, M.: Phys. Rev. Lett. 68, 3259 (1992)2. Einstein, A., Podolsky, B., Rosen, N.: Phys. Rev. 47, 777 (1935)3. Stinespring, W.F.: Proc. Am. Math. Soc. 6, 211 (1955)

76 7 Composite Q-System, Pure State

4. Peres, A.: Phys. Rev. Lett. 77, 1413 (1996)5. Horodecki, M., Horodecki, P., Horodecki, R.: Phys. Lett. A 223, 1 (1996)6. Bell, J.S.: Physics 1, 195 (1964)7. Clauser, J.F., Horne, M.A., Shimony, A., Holt, R.A.: Phys. Rev. Lett. 23, 880 (1969)8. Popescu, S.: Phys. Rev. Lett. 74, 2619 (1995)9. Bennett, C.H., Wiesner, S.J.: Phys. Rev. Lett. 69, 2881 (1992)

10. Bennett, C.H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Phys. Rev.Lett. 70, 1895 (1993)

References 77

Chapter 8All Q-Operations

So far we have learned certain particular q-operations like the unitary transfor-mations, the q-measurements, and the reduction of state. The set of genericq-operations is much larger. We learn the elegant mathematical classification.We discuss its various physical interpretations typically involving the temporaryenlargement of the system by a certain environmental system, subsequent unitaryand measurement operations on the obtained composite system, and an ultimatereduction to the original system.

8.1 Completely Positive Maps

We call a linear mapM positive if it brings the density matrix q of a given systeminto a non-negative matrix whose trace does not exceed 1. In q-theory, we have todefine a subset of positive maps which is the completely positive maps.1 A positivemap M is completely positive if its trivial extension M�I for an arbitrarycomposite system remains positive map. The completely positive maps are alwaysof the following Kraus form (we omit the proof) ( Table 8.1) :

Mq ¼X

n

MnqMyn;X

n

MynMn� I; ð8:1Þ

and the Kraus forms are always completely positive. However, given a completelypositive mapM, its Kraus form is never unique: the Kraus matrices Mn—formingthe ‘‘sandwich’’—can be chosen in many different ways.

There exist non-completely positive maps. Anti-unitary maps are such, like e.g.the reflection T (7.30) on the two-state q-systems. If we try to construct the

1 Completely positive maps were discovered in Stinespring [1]. For their physical elucidation cf.,e.g. Kraus [2].

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79

Kraus representation we fail. Actually, we would get negative sign for some of the‘‘sandwich’’ terms. The reflection T can be written typically into this form:

T q ¼ 12

rqr� 12q: ð8:2Þ

By no means can it be written into the Kraus form (8.1). It is obvious that T is non-completely positive if we recall Sect. 7.2.2 where we saw that the trivial extensionT � I for 2 9 2 dimension was not a positive map.

The non-completely positive maps can not be realized. As we shall see in thenext two sections, the completely positive maps can, at least in principle, berealized in many different ways. What we called q-operations (Sect. 4.2) is nothingelse but the completely positive maps.

8.2 Reduced Dynamics

Consider the composition of the given q-system and a certain, maybe fictitious,environmental q-system E, and introduce a unitary dynamics U on the compositesystem. Let us then reduce the state to the original system, The resultingq-operation is usually non-unitary:

Mq ¼ trE Uðq� qEÞUy� �

: ð8:3Þ

The same q-operation M can be obtained from different unitary dynamics U;moreover, the choice of the environmental system can also be varied. All trace-preserving q-operations can be obtained as reduced dynamics (8.3). The proof canbe done in Kraus representation (8.1). Let us introduce a basis for the environ-mental system E:

n; Ej i; n ¼ 1; 2; . . .; dE: ð8:4Þ

Suppose the initial state of E is pure:

qE ¼ 1; Ej i 1; Eh j: ð8:5Þ

Then the q-operation (8.3) takes the Kraus form

Mq ¼X

n

MnqMyn; ð8:6Þ

Table 8.1 Positive and completely positive maps

The map M : q!Mq is:Positive map if Also completely positive map ifH is Hilbert space HE is arbitrary ‘‘environmental’’ Hilbert spaceq is density matrix on H qbig is density matrix on H�HE

Mq� 0 and tr ðMqÞ� 1 ðM� IÞqbig� 0

80 8 All Q-Operations

if we insert the following Kraus matrices2:

Mn ¼ trE ðI � 1; Ej i n; Eh jÞU� �

� n; Eh jU 1; Ej i: ð8:7Þ

If, the other way around, we know the Kraus form of a trace-preserving q-operationthen the Eq. (8.7) is always solvable for the unitary dynamics U if we assume anenvironmental system E of suitable large dimensions.

8.3 Indirect Measurement

Each ‘‘sandwich’’ term of the Kraus form can be interpreted separately. Supposethat we perform a non-selective q-measurement on the environmental system E.If we do it after the unitary interaction U then it does not influence the reducedstate of the original system. If, however, our measurement is selective then we canarrive at the individual ‘‘sandwich’’ terms of the Kraus form. To see this, we startfrom the basis (8.4) of E, we form the corresponding partition PEn ¼ n; Ej i n; Eh jand extend it trivially for the whole composite system:

Pn ¼ I � PEn: ð8:8Þ

Perform the projective measurement of these physical quantities, after the unitaryinteraction U: The concept is called indirect measurement. We must note of coursethat the measurement of the projectors (8.8) is equivalent with the measurement ofthe projectors PEn on E. Follow the rules of selective measurement (4.13, 4.15).The state of the composite system changes as follows:

Fig. 8.1 Reduced q-dynamics vs. q-operations. We let our system in question interact unitarilywith a (real or fictitious) environmental system. We suppose the uncorrelated initial state q� qE

which the interaction transforms into the q-correlated one Uðq� qEÞUy: Finally we ignore theenvironmental system and we only consider the reduced final state trE½Uðq� qEÞUy�: The unitaryinteraction with the environment has thus induced a non-unitary dynamics for our system, whichwe call reduced dynamics. It corresponds to a completely positive map M: The inverserelationship is also true: A q-operation M (i.e.: a completely positive map) can always berealized by a suitable reduced dynamics

2 To avoid misunderstandings, we note the natural shorthand notation n; Eh jU 1; Ej i stands for amatrix acting on the states q of the original system.

8.2 Reduced Dynamics 81

q� qE !1pn

PnUðq� qEÞUyPn; ð8:9Þ

while the probability distribution of the outcome is

pn ¼ tr PnUðq� qEÞUy� �

: ð8:10Þ

Now we reduce the first equation to the state of the original q-system; the oper-ation trE yields

q! 1pn

MnqMyn; ð8:11Þ

cf. the definition (8.7) of Mn: The second equation takes the equivalent form

pn ¼ tr MynMnq� �

: ð8:12Þ

As we see, each ‘‘sandwich’’ term in the Kraus form (8.1) of our q-operationcorresponds, apart from a normalizing factor, to what comes out from the selectivemeasurement of the environmental system’s basis (8.4). The inverse of therenormalizing factor is always equal to the probability of the given outcome.

We got the key to the interpretation of trace-reducing q-operations. Let us makea selection X of the measurement outcomes and discard the complementary part ofthe q-ensemble. Such q-operation can be written into this form:

MXq ¼Xn2X

MnqMyn;Xn2X

MynMn\I; ð8:13Þ

where this time the summation over the index n is only partial. The obtained stateis not normalized:

Fig. 8.2 Indirect q-measurement vs. selective q-operation. The setup is a refinement of the reduceddynamics. The difference is that after the unitary interaction we do not ignore the state of theenvironment. Rather we perform a projective measurement on it. The ensemble of pre-measurementstate q is selected into sub-ensembles of conditional post-measurement states qn according to theobtained measurement outcomes n. The probability pn coincides with the norm of the unnormalizedconditional state MnqMyn: The sub-ensembles of certain conditional post-measurement statesfqn; n 2 Xg can be re-united contributing to a trace-reducing completely positive map. The inverserelationship is also true: A selective q-operationM (i.e.: a trace-reducing completely positive map)can always be realized by a suitable indirect measurement

82 8 All Q-Operations

trMXq½ � ¼Xn2X

pn: ð8:14Þ

The trace of the state is equal to the total probability of the selected measurementoutcomes n [ X, i.e., it can be less than 1.

The physical interpretation of q-operations has thus been completed. Eachtrace-preserving q-operation can be realized as the reduced dynamics (Sect. 8.2)after the suitable unitary interaction with a suitable environmental system. Eachtrace-compressing q-operation can be realized by a suitable selective indirectmeasurement via the above environmental system. These realizations are neverunique, we can have infinite many choices.

8.4 Non-Projective Measurement Resulting from IndirectMeasurement

Recall the concept of projective (Sect. 4.4.1) and non-projective (Sect. 4.4.2)q-measurements. Projective measurements are also called von Neumann, standard,or ideal measurements. Non-projective ones, also called non-ideal or unsharp, wereformal extensions of projective ones. Now we can easily prove that they are specialcases of indirect measurements. Indeed, (8.11, 8.12) will coincide with (4.20, 4.22)of non-projective measurement in the special case of hermitian Kraus matrices.Then we can always identify the effects by the square of the Kraus matrices:

Pn ¼ M2n ; Mn ¼ Myn ¼ P1=2

n : ð8:15Þ

This way, the mechanism of indirect measurements will underly the concept ofnon-projective measurements. The non-projective measurement of effects Pn canalways be realized by indirect measurements which are just projective mea-surements that happen on a certain, maybe fictitious, environmental systemE q-correlated with the system in question.

Note that there exists an equivalent projective measurement on the uncorrelatedcomposition q� qE of the system and the environment. Only we have to relax thespecial form Pn ¼ I � PnE of the projectors and take the unitarily equivalent set

Pn ¼ UyI � PnEU: ð8:16Þ

This time the projective measurement happens on the composite state q� qE as awhole, not only on the environment:

q� qE !1pn

Pnq� qEPn; ð8:17Þ

pn ¼ tr Pnðq� qEÞ� �

: ð8:18Þ

8.3 Indirect Measurement 83

Nonetheless, the measurement is equivalent, due to unitary equivalence, with theindirect measurement (8.8–8.10) defined previously. The projective measurement ofPn yields the non-projective measurement of the effect Pn on the reduced state q:If Pn yields 1 or 0 then it means that Pn yields 1 or 0 in coincidence. While the formerexpresses sharp information the latter does not. Projective measurement is repeat-able, non-projective is not. When we average over the environmental state we getaveraged data. The effects, too, are averaged version of the projectors:

Pn ¼ trE PnðI � qEÞ� �

: ð8:19Þ

Of course, this expression is equivalent with (8.15).

8.5 Entanglement and LOCC

As we saw earlier in Sect. 7.1.4 of the preceding chapter, local q-dynamics andq-measurements can not create q-correlation. Having learned the class of generalq-operations (Sect. 8.1), we can now formulate the precise statement. In thepostulated situation, Alice and Bob own their separate local systems far from eachother and the remote systems form a composite q-system together. Alice and Bobare allowed to perform local operations (LO). When they do so independently thenthe composite system changes this way:

qAB �! ðMA �MBÞqAB: ð8:20Þ

Usually we suppose classical communication (CC) as well. Alice and Bob caninform each other about the setups and the outcomes of their local operations sothat they can condition their further LOs on the received classical information.Q-correlation (entanglement), if it is not already present, can never be produced byLOCC. To see this, suppose the initial state is uncorrelated ðqA � qBÞ and Aliceperforms her LO MA selectively, see Eqs. (8.11) and (8.12):

qA !1

pAnMAnqAMyAn � qAn: ð8:21Þ

Suppose that, using CC, Alice tells Bob of the outcome n: Bob can make his LO

MB dependent on n. Let MðnÞB stand for Bob’s LO. The selective change of the

composite state is this:

qA � qB ! qAn �MðnÞB qB; ð8:22Þ

and the non-selective post-LOCC state becomes:

qA � qB !X

n

pAnqAn �MðnÞB qB: ð8:23Þ

84 8 All Q-Operations

This is a separable state containing classical correlations only. The proof extendstrivially to the case when the initial state is already a mixture of uncorrelated statesðqAk � qBkÞ: Any LOCC will preserve such a separable structure.

8.6 Open Q-System: Master Equation

When a q-system is continuously interacting with certain environmental q-systemor q-systems then it is called open q-system. A subsystem of a closed systemmakes a typical open system. The open system dynamics is reduced dynamics andis thus irreversible. There exists an idealized (so-called Markovian) class ofenvironmental interactions where the state would still satisfy an equation similar tovon Neumann’s (4.5). The Markovian master equation takes this form:

dqdt¼ � i

�h½H; q� þ non-Hamiltonian terms � Lq: ð8:24Þ

It should generate a completely positive map MðtÞ ¼ expðtLÞ: This conditionimplies that the master equation can always be written into Lindblad form [3, 4](although we do not prove it here):

dqdt� Lq ¼ � i

�h½H; q� þ

Xn

LnqLyn �12fLynLn; qg

� �: ð8:25Þ

The supermatrix L is called the Lindblad generator. Given a master equation by L;its Lindblad form is never unique: the Lindblad matrices Ln—forming thenon-Hamiltonian term—can be chosen in many different ways. The physicalinterpretation of Lindblad matrices is related to the form of the interaction betweenthe open system and its environment. Choice of Lindblad matrices of a given opensystem is non-unique because the same reduced dynamics (Sect. 8.2) of the opensystem can be obtained by various interactions with various environments.

8.7 Q-Channels

A typical theoretical application of q-operationsM is the concept of a q-channel.A q-channel serves to communicate an arbitrary q-state q from one location toanother, say, from Alice to Bob like in the protocols of the q-banknote (Sect. 6.4.1),q-key distribution (Sect. 6.4.2), or superdense coding (Sect. 7.3.1). In the idealcase, these protocols use noiseless q-channels: the output state coincides with theinput one. In general, however, the noisy q-channel will distort the state, it performsa certain q-operationM on the input. Terminologically, we identify the operationM and the q-channel. Q-channels are categorized according to the distortions they

8.5 Entanglement and LOCC 85

cause. Elementary q-channels are those communicating single independent qubits.The depolarizing channel is this:

q!Mq ¼ ð1� wÞqþ wI

2; ð8:26Þ

it damps the polarization of a qubit isotropically by a factor 1 - w. The dephasingchannel takes this form:

q!Mq ¼ ð1� wÞqþ wrzqrz; ð8:27Þ

which decreases the off-diagonal elements of q in the computational basis by afactor 1 - w.

8.8 Problems, Exercises

8.1 All q-operations are reductions of unitary dynamics. Consider a trace-preserving q-operation M: It is equivalent with a certain reduced dynamicswhere the system interacts unitarily with a ‘‘fictitious’’ environmental systemof suitably large dimension dE. Let us prove this equivalence starting from theKraus representation of M and show that it suffices if we choose dE tocoincide with the number of the Kraus matrices Mn: Method: introduce thebases kj i; n; Ej i for the system and the environment, respectively, and observethat the states

Pn Mn kj i � n; Ej i form an orthonormal set.

8.2 Non-projective effect as averaged projection. Let us write the uncorrelatedcomposite state of the system and the environmental system as qqE: We canmany times omit the notation of � as well as I without risk of misunder-standing. In this example we use such convenience. Suppose we perform aprojective measurement of the partition fPng of the composite system. If weignore the post-measurement state of the environment, we are left with a givennon-projective measurement of the positive decomposition fPng on thesystem alone. Let us prove that Pn ¼ trEðPnqEÞ:

8.3 Q-operation as supermatrix. Consider an operation M which transformsdensity matrices into density matrices. In a given basis, the density matricesare represented by their components qkl ¼ kh jq lj i and the operation is rep-resented by the components of a supermatrix of four indices:

qkl �!Xk0l0

Mklk0l0qk0l0 :

Let us express the supermatrix M by utilizing the Kraus representation.8.4 Environmental decoherence, time-continuous depolarization. If we are inter-

ested in the complete isolation of our q-system then all environmental

86 8 All Q-Operations

interactions are considered as parasitic. They tend to destroy superpositionsand entanglements within our system. They will paralyze, in particular, q-protocols and q-algorithms that take long time enough to accumulate theabove environmental decoherence. A simplest model of an uncontrollableenvironmental interaction is the time-continuous depolarization of the qubit.Let us show that the master equation

dqdt¼ � 1

4sðrqr� 3qÞ

is of the Lindblad form and it describes isotropic depolarization. Derive theequation of motion for the polarization vector s:

8.5 Kraus representation of depolarization. Let us construct Kraus representationfor the depolarization channel. Method: exploit the isotropy of the channel.

References

1. Stinespring, W.F.: Proc. Am. Math. Soc. 6, 211 (1955)2. Kraus, K.: States, Effects, and Operations: Fundamental Notions of Quantum Theory.

Springer, Berlin (1983)3. Lindblad, G.: Commun. Math. Phys. 48, 199 (1976)4. Gorini, V., Kossakowski, A., Sudarshan, E.C.G.: J. Math. Phys. 17, 821 (1976)

8.8 Problems, Exercises 87

Chapter 9Classical Information Theory

Shannon entropy is the key-notion of classical information. It provides thestatistical measure of information associated with states q. Since dynamicalaspects shall not be treated at all, we would just talk about probability distributionsp instead of physical states q. For comparability with Q-information theory ofChap. 10, however, we keep talking about states q of classical systems. Typically,we use heuristic proofs though corner stones of the exact derivations will fairly beindicated.

9.1 Shannon Entropy, Mathematical Properties

Shannon entropy in statistical physics used to be introduced for a classical stateq(x) defined on continuous phase space {x}. For convenience of classical infor-mation theory, however, we shall use states q(x) defined on discrete space {x}, cf.Sect. 2.7.

Shannon entropy of a given state will be defined as follows:

SðqÞ ¼ � log qh i ¼ �X

x

qðxÞ log qðxÞ: ð9:1Þ

The Shannon entropy takes its minimum value 0 if the state is pure, sayq(x) = dx0. It takes its maximum value if the state is random: q(x) = const.

Mixing increases the entropy. The entropy of a weighted mixture is greater than(or equal to) the weighted sum of the constituents’ entropies (concavity):

Sðw1q1 þ w2q2Þ�w1Sðq1Þ þ w2Sðq2Þ ; w1 þ w2 ¼ 1: ð9:2Þ

Correlation decreases the entropy. The entropy of a bipartite composite state is lessthan (or equal to) the sum of the entropies of the two reduced states. The equalityonly holds when the two subsystems are uncorrelated (subadditivity):

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89

SðqABÞ� SðqAÞ þ SðqBÞ: ð9:3Þ

Reduction decreases the entropy. The entropy of reduced states is less than (orequal to) the entropy of the composite state:

SðqABÞ� SðqAÞ; SðqBÞ: ð9:4ÞThe relative entropy can be defined for two states q, q0 that belong to the samesystem:

Sðqkq0Þ ¼ �SðqÞ �X

x

qðxÞ log q0ðxÞ: ð9:5Þ

The relative entropy is zero if q = q0 and it is positive otherwise.

9.2 Messages

According to the abstraction of classical information, a finite sequence of letterstaken from an alphabet constitutes a message. Messages can be used to store orcommunicate the encoded data. The simplest alphabet is binary, it has two lettersusually associated with 0 and 1, respectively. The binary letters are also called bits.Binary messages are sequences of binary digits and can be thought of integers inbinary representation. The unit of classical information measure is one random bit,or one bit in short (meaning always one random bit). The information of a randomn-letter binary message is thus n bits. If the alphabet is not binary the messages canstill be faithfully translated into binary messages. One random letter of the K-letteralphabet can, on average, be converted into log K random binary letters (bits), i.e.,into a log K-letter long random binary message. A shorter binary message couldnot encode the original letter faithfully. Therefore we say that a random letter of aK-letter alphabet contains log K bits information.

If the letter is not random it can be compressed faithfully into less than logK random bits. The original message is redundant and the measure of informationis less than nlog K bits. In general, we define the information of a message by thelength of the corresponding shortest faithfully compressed binary message.According to Shannon’s theory, this shortest length is nS(q) in the limit n ? ?[1]. We outline the proof.

9.3 Data Compression

Consider an n-letter long message of a K-letter alphabet {a, b, c, . . . , z}:

x1x2; . . .; xn; ð9:6Þ

and suppose that the letters independently follow the same a priori probabilitydistribution:

90 9 Classical Information Theory

qðaÞ ¼ p1; qðbÞ ¼ p2; qðcÞ ¼ p3; . . .; qðzÞ ¼ pK : ð9:7Þ

Physically, each letter is represented by a discrete K-state classical system and thestatistics of n-letter long messages is represented by the collective state

qðx1Þqðx2Þ; . . .; qðxnÞ � q�nðx1; x2; . . .; xnÞ: ð9:8Þ

The letters are random if q = const, otherwise the data contained in the abovecollective state can be compressed.

Indeed, many combinatorically possible messages are totally unlikely andbecome negligible if n ? ?. Only a portion of the combinatorically possiblemessages will dominate statistically. These are called typical messages. In them,the relative frequency of each letter is identical to its a priori probability (9.7)! If ina very long message this rule were violated for any letter then the message wouldbecome atypical and thus statistically irrelevant. Therefore in typical messages theletters of the alphabet occur with frequencies

np1; np2; . . .; npK : ð9:9Þ

The typical messages can only differ from each other by the order of the letters!All typical messages have identical a priori probability. It is thus sufficient if wecalculate the number of the typical messages, their probability distribution is evenand equals the inverse of their number.

The number of the typical messages equals the number of polynomial combi-nations of n elements satisfying the frequencies (9.9):

No. of typical messages ¼ n!

ðnp1Þ!ðnp2Þ!. . .ðnpKÞ!: ð9:10Þ

If n is large, Stirling approximation

log n! ¼ n log n� n log eþ Oðlog nÞ ð9:11Þ

applies to all factorials, yielding

logn!

ðnp1Þ!ðnp2Þ!. . .ðnpKÞ!� �n

XK

k¼1

pk log pk: ð9:12Þ

The r.h.s. is just n-times the Shannon entropy of a single (non-random) letter.Accordingly, the number of typical messages reads

No. of typical messages ¼ 2nSðqÞ: ð9:13Þ

Their distribution is even. The collective state q9n of the original messages can,information theoretically, be replaced by the even distribution of the typicalmessages:

9.3 Data Compression 91

q�n �! 1

2nSðqÞXn; ð9:14Þ

where Xn(x1, x2, . . . , xn) is the indicator function which is 1 for the 2nS(q) typicalmessages and zero otherwise. The typical ones of the original redundant messages(9.6) can therefore be numbered by nS(q) binary digits constituting nS(q)-bitrandom messages. These binary messages are not redundant, cannot be furthercompressed faithfully. We have in such a way obtained the shortest faithful binarycompression. Each letter of the original message has been faithfully compressedfrom log K bits into S(q) bits.

This data compression theorem is the information theoretic interpretation of theShannon entropy. The theorem as well as its interpretation are asymptotic, validfor one letter in infinite long redundant messages. This explains the success of theabove heuristic proof. It must be obvious that for finite n the frequencies of theletters would fluctuate around the asymptotic values (9.9), the set of typicalmessages might increase. Nevertheless, the exact proof would confirm that theabove heuristic one has yielded the correct asymptotic number of the statisticallyrelevant messages.

9.4 Mutual Information

Let us introduce the notation X = {x, q(x)} for a single-letter message where q(x)is the a priori distribution of single letters. Shannon entropy S(q) has beenattributed to the distribution q(x) of single letters. For terminological and nota-tional conveniences we say that this is the entropy of the (single-letter) messageX and we introduce the alternative notation H(X) for it:

HðXÞ ¼ SðqÞ: ð9:15Þ

The Shannon entropy measures the rate of our a priori ignorance concerning longmessages. We must learn H(X) bits per letter to eliminate our ignorance perfectly.

Suppose two separate messages X and Y are letter-wise correlated according toa certain distribution q(x, y). Let H(X, Y) stand for the joint Shannon entropy. Wedefine the mutual information of the two messages:

IðX :YÞ ¼ HðXÞ þ HðYÞ � HðX; YÞ ¼ logqðx; yÞ

qðxÞqðyÞ

� �: ð9:16Þ

This is zero if the two messages X and Y are uncorrelated. Otherwise it is positive.The mutual information I(X : Y) gives the number of bits, per composite letter(x, y), that we can spare when we compress the joint message (X, Y) instead ofcompressing X and Y separately.

The mutual information is related to the conditional entropy. While I(X : Y) issymmetric function of the two messages the conditional entropy is not. The

92 9 Classical Information Theory

conditional entropy of the message X is built upon the conditional probabilityqðxjyÞ = q(x, y)=q(y):

HðXjYÞ ¼ �hlog qðxjyÞi ¼ HðX; YÞ � HðYÞ: ð9:17Þ

Conditional entropy means the minimum number of bits (per letter) needed toencode the message X if the message Y is perfectly known. H(XjY) = H(X) if thetwo messages are uncorrelated, and H(XjY) = 0 if they are perfectly correlated.The mutual information and the two conditional entropies are simply related:

IðX :YÞ ¼ HðXÞ � HðXjYÞ ¼ HðYÞ � HðYjXÞ: ð9:18Þ

9.5 Channel Capacity

Let X be the input message of a noisy channel and let Y be its output message. Thechannel is characterized by the transfer probability qðxjyÞ. Then the joint distri-bution of the input and output letters becomes

qðx; yÞ ¼ qðyjxÞqðxÞ: ð9:19Þ

In fact, we can learn the output message Y and, from it, we have to estimate theinput message X. According to Bayes rule, we write down the aposteriori distri-bution of the input letter conditioned on the received output letter:

qðxjyÞ ¼ qðx; yÞqðyÞ ¼

qðyjxÞqðxÞPx0 qðyjx0Þqðx0Þ

: ð9:20Þ

This becomes proportional to the transfer probability qðxjyÞ of the channel ifq(x) = const, i.e., when the input message is random:

qðxjyÞ ¼ qðyjxÞPx0 qðyjx0Þ

: ð9:21Þ

The mutual information I(X:Y) (9.18) indicates how many bits of the inputmessage X can faithfully be decoded from the perfect knowledge of one outputletter. If the channel is noiseless, qðxjyÞ = dxy, then H(X) = H(Y) =H(X, Y) = I(X : Y) meaning that the output yields the input perfectly. In thecontrary case, the channel noise makes the output message random: qðyjxÞ= const. Then HðXÞ ¼ HðXjYÞ and I(X:Y) = 0 so we cannot estimate the inputat all.

In the intermediate cases we obtain partial information on the input. Out of theH(X) bits information per input letter, the output still contains I(X:Y) bits infor-mation per letter. Those I(X:Y) bits are in general not sufficient to faithfullyestimate the input letter itself. Yet, the I(X:Y) bits represent faithful partialinformation on the input message. The given channel communicates I(X:Y) bits/

9.4 Mutual Information 93

letter faithful information. This rate depends on the transfer probability q(y|x) ofthe channel and on the distribution q(x) of the input letters, see also (9.18, 9.19).By an optimum choice of q(x) we can maximize the the rate of faithful commu-nication. This rate is called the channel capacity

C � maxfqðxÞg

IðX :YÞ; ð9:22Þ

which is the function of the transfer probability q(y|x) alone.

9.6 Optimal Codes

Shannon entropy H(X) of a message X has been interpreted as the bit-length of theshortest faithful code per letter. Our heuristic proof 9.3 could be made exact. First,one has to construct the block-code of n-letter long messages in such a way that, inthe asymptotic limit n ? ? the code length is * nH(X) bits. Note that the block-code cannot be decomposed into letter-wise codes. Second, one has to prove thatthis code is faithful, i.e., the decoding yields the original message at error ratewhich goes to zero for n ? ?. Third, one has to prove that further data com-pression would necessarily violate faithfulness. Shannon has done the exact proof,that is his noiseless coding theorem.

In a similar way could we make the heuristic proof of channel capacity 9.5exact. We consider nC-bit random messages to communicate through the channelfaithfully. First we encode them properly into the *n-letter long input messagesX of the channel. Note that C B H(X), i.e., the chosen code must be redundant ifthe channel is noisy. Second, we prove that, in the limit n ? ?, decoding theoutput message Y yields the initial nC bits faithfully. Third, we prove that for thegiven channel any higher rate of faithful communication is impossible. This isShannon’s noisy code theorem.

9.7 Cryptography and Information Theory

The cryptography situation 6.4.2 can be interpreted in terms of classical informa-tion theory. There are three parties: Alice, Bob, and Eve. Alice generates therandom key X, Bob estimates a certain key Y which is a noisy version of X, whileEve steals a key Z which is another noisy version of X. The quality of a given key-distribution protocol, whether classical or quantum, depends on the three mutualinformations I(X : Z), I(Y : Z) and I(X : Y). The success and faithfulness of key-dis-tribution is measured by I(X : Y), the loss of security is measured by I(X : Z) andI(Y : Z). In the ideal case I(X : Y) = H(X) = H(Y) and I(X : Z) = I(Y : Z) = 0. Theultimate security of q-key protocols against unnoticed evedropping is based, amongothers, on the analysis of these mutual informations [2].

94 9 Classical Information Theory

9.8 Entropically Irreversible Operations

We introduced the qualitative notion of irreversibility on examples, e.g., of thereduced dynamicsMðtÞ in Sect. 2.5 where it meant just the non-invertibility of theoperationMðtÞ: The informatical concept of irreversibility is more restrictive thanthe concept of non-invertibility. It assumes information loss or, in alternativeterminology: entropy increase. In typical reduced dynamics, e.g., the entropy forsome initial states is increasing while for some other ones it is decreasing. Below,we are going to characterize the general class of entropy increasing operations.

A non-invertible operation M is (entropically) irreversible if it increases orpreserves the entropy:

SðMqÞ� SðqÞ ; ð9:23Þ

for all states q. We shall show that a non-invertible M is irreversible if it leavesthe totally mixed state q0(x) = const invariant: Mq0 ¼ q0:

On a discrete phase space, the class of invertible operations is narrow: it cor-responds to the permutations of the discrete points {x} of the state space. All otherdiscrete classical operationsM are non-invertible. We are going to show thatMincreases (or at least preserves) the entropy for all q if Mq0 ¼ q0: The generalform of an operation q! q0 � Mq reads: q0ðyÞ ¼

Px qðyjxÞqðxÞ where q(yjx) is

the non-negative transfer function, satisfyingP

y qðyjxÞ ¼ 1 for all x. From the

condition Mq0 ¼ q0 it follows that our transfer function is normalized in x aswell:

Px qðyjxÞ ¼ 1 for all y. Such transfer function is also called doubly-sto-

chastic. We can always write

Sðq0Þ � SðqÞ ¼Xx;y

qðxÞqðyjxÞ logqðxÞq0ðyÞ: ð9:24Þ

We invoke the inequality ln k[ 1� k�1 valid for k = 1 and apply it to k = q=q0.This yields

Sðq0Þ � SðqÞ� 1ln 2

Xx;y

qðyjxÞ½qðxÞ � q0ðyÞ� ¼ 0: ð9:25Þ

In general, the strict inequality holds for q = q0 unless the support of q is smallerthan the whole state space and the restriction of the non-invertible M for thatsupport becomes invertible.

Let us see the simplest example. If the channel yields totally random outputmessage always then it is entropically irreversible, it will increase the entropy forall possible messages apart from the marginal cases mentioned above. As a con-trary example, consider a ‘‘broken’’ channel whose output message is a permanent0 for any input whatsoever. The channel operation is obviously non-invertible butit is not entropically irreversible.

9.8 Entropically Irreversible Operations 95

9.9 Problems, Exercises

9.1 Positivity of relative entropy. Using the inequality ln k [ 1� k�1; let us provethe positivity of the relative entropy.

9.2 Concavity of entropy. Why and how does mixing increase the entropy? Let usargue information theoretically. Method: start from two messages of differentlengths proportional to w1 and w2, and of different single-letter distributionsq1, q2, respectively.

9.3 Subadditivity of entropy. Using the positivity of the relative entropy, let usprove the subadditivity of the entropy.

9.4 Coarse graining increases entropy. Consider a given classical state, i.e.probability distribution q(x) defined on the period [0,1] represented to aprecision of k binary digits: x : 0.x1x2, . . . ,xk. Let us introduce the coarse-grained state

~qð~xÞ ¼X

xk¼0;1

qðxÞ;

defined on the same state space [0,1] to a precision of one digit less, i.e., weintroduced ~x � 0:x1x2; . . .; xk�1: Let us prove that coarse graining increasesthe entropy: Sð~qÞ� SðqÞ: Method: use the fact that state reduction increasesthe entropy.

References

1. Shannon, C.E.: Bell Syst. Tech. J. 27, 379–623 (1948)2. Bennett, C.H.: Phys. Rev. Lett. 68, 3121 (1992)

96 9 Classical Information Theory

Chapter 10Q-Information Theory

The q-information theory is similar to the classical one. The carrier of informationis a q-system and this causes differences as well. A particular difference is thatqubits carry information in double sense. First, we can encode classical bits intoqubits. Second, unknown qubits carry information which is hidden and protected—as a consequence of the universal limitations of single q-state determination,cloning, or distinguishability.

10.1 Von Neumann Entropy, Mathematical Properties

Von Neumann entropy, or q-entropy, of a given q-state q will be defined as follows:

SðqÞ ¼ � log qh i ¼ �tr q log qð Þ: ð10:1Þ

Obviously, the q-entropy is unitary invariant:

SðqÞ ¼ SðUqUyÞ: ð10:2Þ

The von Neumann entropy takes its minimum value zero if the state is pure, i.e.:SðjwihwjÞ ¼ 0. It takes its maximum value on the maximally mixed state whenSðI=dÞ ¼ log d:

Let us construct the spectral expansion of the q-state q:

q ¼X

k

qkjukihukj: ð10:3Þ

The eigenvalues qk of q form a probability distribution and its Shannon entropyequals the von Neumann entropy of the q-state q:

L. Diósi, A Short Course in Quantum Information Theory,Lecture Notes in Physics, 827, DOI: 10.1007/978-3-642-16117-9_10,� Springer-Verlag Berlin Heidelberg 2011

97

SðqÞ ¼ �X

k

qk log qk ¼ SðqÞ: ð10:4Þ

Mixing increases the q-entropy. The q-entropy of a weighted mixture is greaterthan (or equal to) the weighted sum of the constituents’ q-entropies (concavity):

Sðw1q1 þ w2q2Þ�w1Sðq1Þ þ w2Sðq2Þ; w1 þ w2 ¼ 1: ð10:5Þ

Correlation decreases the q-entropy. The q-entropy of a bipartite compositeq-system is less than (or equal to) the sum of the q-entropies of the two reducedq-states. The equality only holds when the two q-subsystems are uncorrelated(subadditivity):

SðqABÞ� SðqAÞ þ SðqBÞ: ð10:6Þ

Reduction may change q-entropy in both directions, contrary to classical entropy.The difference between the q-entropies of the two reduced q-states cannot begreater than the entropy of the bipartite composite q-state (triangle inequality):

SðqABÞ� jSðqAÞ � SðqBÞj: ð10:7Þ

This is a genuine q-feature. The Shannon entropy of a classical sub-system isnever greater than the entropy of the composite system (9.4):

SðqABÞ� SðqAÞ; SðqBÞ: ð10:8Þ

Hence the information content of a classical sub-system is smaller than (or equalto) the composite system’s entropy. For q-systems this shall not be true. Typically,a pure entangled q-state has zero q-entropy while its subsystems are in mixedreduced states of positive q-entropy. Contrary to quantum, the classical purecomposite states are so trivial that their subsystems remain pure.

The relative q-entropy can be defined for two states q; q0 that belong to thesame q-system:

Sðqkq0Þ ¼ �SðqÞ � trðq log q0Þ: ð10:9Þ

The relative entropy is zero if q ¼ q0 and it is positive otherwise. The inequalitySðq0kqÞ� 0 is also called as the Klein inequality.

10.2 Messages

Consider a classical message X = {x, q(x)} which we encode letter-wise intocertain, not necessarily orthogonal, pure states of a given d-state q-system:

letter: x1 x2 . . . xn

q-code: jx1i jx2i . . . jxniprobability: qðx1Þ qðx2Þ . . . qðxnÞ

ð10:10Þ

98 10 Q-Information Theory

The sequence jx1i; jx2i; . . .; jxni is a q-message of length n. Q-messages can be usedto store or communicate the classical data encoded into q-states. We shall make anultimate q-measurement on each q-code jxi in order to infer the encoded classicalinformation to the best possible extent. A single pure state jxi can be unitarilytransformed into log d qubits, and the above q-message can be transformed unitarilyinto a binary q-message of n log d qubits. The unit of q-information measure is onerandom qubit, or one qubit in short (meaning always one random qubit). Theinformation of a random n-qubit q-message is thus n qubits. We can decide if aq-message is random by constructing the density matrix of the 1-letter q-message:

q ¼X

x

qðxÞjxihxj: ð10:11Þ

The q-message is random if this density matrix is totally mixed: q ¼ I=d: Then then-letter q-message is unitarily equivalent with n log d random qubits. If theq-message is not random it can be compressed faithfully into less than n logd random qubits. The original q-message is redundant and the measure ofq-information is less than n log d qubits. In general, we define the information of aq-message by the length of the corresponding shortest faithfully compressedbinary q-message. According to Schumacher’s theory [1], this shortest length isnSðqÞ in the limit n ? ?. We outline the proof which resembles the proof of theclassical case in Sect. 9.3.

10.3 Data Compression

From the viewpoint of q-theory, the q-message jx1i; jx2i; . . .; jxni is a multiplecomposite pure state in a dn-dimensional Hilbert space:

jWi ¼ jx1i � jx2i � � � � � jxni: ð10:12Þ

This state can be realized by n independent d-state q-systems. The average of thisstate over all classical n-letter messages x1; x2; . . .; xn will be the following col-lective (4.53) mixed state:

Xfxg

qðx1Þ. . .qðxnÞjWihWj ¼ q� q� � � � � q � q�n: ð10:13Þ

If q 6¼ I=d; the q-message jWi is not random; it is redundant and the measure ofq-information is less than n log d qubits. Indeed, many directions jWi in the Hilbertspace are totally unlikely to occur and become negligible if n ! 1. Only asubspace of the whole dn-dimensional Hilbert space will dominate statistically.This subspace is called the typical subspace. We shall represent it by a hermitian

projector Xn which will be determined for asymptotically large n. It can be shownthat the maximum faithful compression of the pure state messages jWi corresponds

10.2 Messages 99

to the maximum faithful compression of the average message q�n. The taskreduces to the maximum faithful compression of the collective state q�n:

Of course, classical compression of the initial messages x1; x2; . . .; xn may notlead to an optimum compression of the encoded q-messages. Yet, as we shallargue, this becomes true provided the q-codes jxi are orthogonal. We shall tem-porarily assume such orthogonal codes and utilize Shannon’s classical datacompression theorem. Note then that SðqÞ ¼ SðqÞ: Furthermore, we can perfectlydiscriminate the different q-messages (10.12) because they are orthogonal to eachother so that we can simply use the corresponding projective q-measurement.Consequently, the best q-compression turns out to be equivalent with the bestclassical data compression learnt in Sect. 9.3. We only retain those q-messages(10.12) which correspond to the typical classical messages. We learned that theirnumber is 2nS(q). The corresponding composite vectors jWi span a 2nSðqÞ ¼ 2nSðqÞ -dimensional typical subspace. Its orthogonal complement in the dn-dimensionalHilbert space is called the atypical subspace. Since the probability of each clas-sical typical message is just 2�nSðqÞ; the faithful compression of the averagemessage yields the following new density matrix:

q�n �! 1

2nSðqÞXn; ð10:14Þ

where Xn is the hermitian projector onto the 2nSðqÞ-dimensional typical subspace.Q-data compression means that the original redundant q-messages (10.12) will be

orthogonally projected by Xn into the typical subspace. The typical subspace isunitary equivalent with nSðqÞ qubits. The compressed state corresponds to nSðqÞ -qubit random q-messages. These binary q-messages are not redundant, cannot befurther compressed faithfully. We have in such a way obtained the shortest faithfulbinary compression. Each q-letter jxi of the original q-message has been faithfullycompressed from log d qubits into SðqÞ qubits.

It is time to come back to the original task (10.10) and allow for non-orthogonalq-codes jxi as well. It can be shown that the desired faithful q-data compressiondepends on the 1-letter density matrix q only. Therefore the faithful q-data com-

pression takes always the form (10.14). The projector Xn will be calculated asbefore: for the given q; there can always be constructed a hypothetical q-message

(10.10) with orthogonal q-codes. Construction of Xn is thus straightforward if we

know q: The q-data compression will be realized by the projection Xn jWi into thetypical subspace of q�n: The compressed q-code becomes a 2nSðqÞ-dimensionalstatevector and these statevectors occur at random within the typical subspace. Suchstatevectors are equivalently described by nSðqÞ qubits which are also random. Thisway we have outlined the proof of Schumacher’s q-data compression theorem: theclassical message x1; x2; . . ., when encoded into the non-orthogonal pure statesjx1i; jx2i; . . .; carries SðqÞ qubit q-information per letter for asymptotically longmessages, where q is the density matrix of the single letter q-message and SðqÞ is itsvon Neumann entropy.

100 10 Q-Information Theory

Von Neumann entropy SðqÞ has thus been interpreted as the qubit-length per letterof the shortest faithful q-code of the message whose 1-letter average density matrix isq. The above heuristic proof could be made exact. First, one has to construct theq-block-code of n-letter long messages in such a way that, in the asymptotic limitn ? ? the q-code length is � nSðqÞ qubits. Note that the q-block-code cannot bedecomposed into letter-wise codes. It must be a multiply entangled pure state of all� nSðqÞ qubits. Second, one has to prove that the compression is faithful, i.e., it doesnot decrease the accessible information 10.4. Third, one has to prove that further datacompression would necessarily violate faithfulness. Schumacher has done the exactproof, that is his noiseless q-coding theorem.

10.4 Accessible Q-Information

In the previous section, certain classical messages X = {x, q(x)} have been letter-wise encoded into pure q-states. More generally, we can encode letters into mixedq-states, too:

letter: x1 x2 . . . xn

q-code: qx1qx2

. . . qxn

probability: qðx1Þ qðx2Þ . . . qðxnÞð10:15Þ

The density matrix of the averaged 1-letter code reads

q ¼X

x

qðxÞqx: ð10:16Þ

So far we have not discussed how much information is accessible regarding theoriginal classical message X, by measuring the q-code. Now we apply a non-projective q-measurement 4.4.2 to each q-code qx: Given the set of effects fPyg;the measurement outcome y will be considered a letter of another classical mes-sage Y whereas the probabilities of letters y are conditioned on the original letter x:

qðyjxÞ ¼ tr Pyqx

� �: ð10:17Þ

Formally, this is the transfer function of a noisy classical channel. From the outputmessage Y we can letter-wise estimate the input message X. For statistics of longmessages, the faithful average information carried by one letter y is called theinformation gain and measured by the mutual information 9.4:

Igain ¼ IðX :YÞ ¼ HðXÞ � HðXjYÞ: ð10:18Þ

For the noisy classical channel itself, this was considered the communicatedamount of information and we searched its maximum by optimizing the inputmessage, recall Sect. 9.4. This time, however, we take both the input message andits q-code granted and search the maximum of I(X:Y) by optimizing the q-effectsfPyg of our non-projective q-measurement:

10.3 Data Compression 101

A � maxfPyg

IðX :YÞ: ð10:19Þ

This A is called the accessible information referred to the classical message X,from its q-codes (10.15). In the marginal case, the q-codes are orthogonal purestates. Then the optimum q-effects are orthogonal projectors and their projectiveq-measurement yields complete information on the original message: A = H(X).The general case is much more difficult.

10.5 Entanglement: The Resource of Q-Communication

Q-communication between Alice and Bob needs a q-channel typically. Lacking aq-channel, Alice and Bob can cope with teleportation 7.3.2 which needs sharedentangled states. One shared Bell state allows teleportation of one qubit, n Bellstates allow teleportation of n qubits. We say: entanglement is the resource ofq-communication. And vica versa, q-communication is a resource of sharedentanglement. Alice and Bob can share n Bell states if they can communicaten qubits through their q-channel.

Accordingly, in the lack of q-communication the amount of (shared) entan-glement is constant or at least not increasing, cf. Sect. 8.5. Alice and Bob cantransform their entanglement into various forms but the amount of entanglementremains the same all the time. In q-information theory we consider the so-calledLOCC situation where Alice and Bob can do any local operation on theirrespective local systems, and they can also communicate classically: but theycannot make q-communication. An important field of q-information theory isstudying various q-state manipulations at LOCC constraints.

The prototype of the entangled state is a maximally entangled Bell state

jUþABi ¼j0; Ai � j0; Bi þ j1; Ai � j1; Biffiffiffi

2p � j00i þ j11iffiffiffi

2p : ð10:20Þ

This is the unit of entanglement measure, too. For it: E = 1. We say that one qubitis Alice’s local qubit, the other qubit is Bob’s. When they share k such Bell stateswe can say that they share entanglement E = k. If they share k pairs of qubits thatare only partially entangled, e.g.:

jWABi ¼ cos h j00i þ sin h j11i; ð10:21Þ

then the entanglement per pair is smaller then E = 1, Alice and Bob share lessentanglement than E = k. We made a reasonable choice earlier as to the measureof the partial entanglement. Accordingly, we calculate the reduced states of Aliceand/or Bob:

qA ¼ qB ¼ cos2 h j0ih0j þ sin2 h j1ih1j: ð10:22Þ

102 10 Q-Information Theory

The partially entangled state (10.21) will, by our heuristic definition (7.13), pos-sess an entanglement measured by the von Neumann entropy of the above reducedstates:

E ¼ S ¼ �ðcos2 hÞ logðcos2 hÞ � ðsin2 hÞðlog sin2 hÞ: ð10:23Þ

This heuristic definition fits to the minimum (E = 0) and maximum (E = 1) entan-glements. The interpretation of the intermediate values is only possible if we learnthe reversible cycle of entanglement distillation–dilution. We are going to prove inthe next section that, at LOCC constraints, a large number n of partially entangledstates jWABi can be distilled into k = Sn maximally entangled states jUþABi: ThisLOCC operation is reversible: a large number k of maximally entangled states jUþABican be diluted into k/S partially entangled states jWABi; as we see in the section afterthe next. Then it is crucial to understand that neither the efficiency of distillation northe efficiency of dilution can be further improved. Why? Because we would thenmultiply the number of Bell states in a single distillation–dilution or dilution–distillation cycle. And this is impossible by LOCC, cf. Sect. 8.5.

In summary: the entanglement E of a composite pure state is interpreted by therate of the maximum number of distillable Bell states, and this, as we are going toprove, is indeed equivalent to the heuristically found definition E = S.

10.6 Entanglement Concentration (Distillation)

Suppose Alice and Bob share n copies of partially entangled state jWABi withentanglement E = S; let n be large. We show that, using LOCC only, Alice andBob can distill their states into k*nE = nS maximally entangled Bell states:

jWABi�n �! jUþABi�k: ð10:24Þ

To this end, Alice performs a smart collective q-measurement on her n qubits. Themeasured physical quantity will be the overall polarization in direction z. In thecomputational representation, this is equivalent with the collective measurement ofthe sum m of the binary physical quantities x (5.2) for Alice’s n qubits, where:

m ¼ x� I�ðn�1Þ þ permutations

� x� I�ðn�1Þ þ I � x� I

�ðn�2Þ þ � � � þ I�ðn�1Þ � x:

ð10:25Þ

If the outcome is a given integer m then the rules of projective q-measurementimpose the following state change:

10.5 Entanglement: The Resource of q-Communication 103

jWABi�n �! j11i�m � j00i�ðn�mÞ þ permutationsffiffiffiffiffiffiffiffiffiffiffiffiffinm

� �s : ð10:26Þ

where the permutations include all states where m qubits are x = 1 and the othern - m qubits are x = 0. The most probable value of measurement outcome m isapproximately this:

n sin2 h; ð10:27Þand by that shall we approximate Alice’s measurement outcome. If n ? ?, theStirling formula (9.11) applies to the average number d of permutations on ther.h.s. of (10.26):

d ¼ nm

� � n

n sin2 h

� � 2nS ¼ 2nE; ð10:28Þ

where S just turns out to be the von Neumann entropy (10.23). It was adopted asthe heuristic entanglement E of the state jWABi: Hence Alice’s collective mea-surement (10.26) has led approximately to the following entangled q-state:

j11i�m � j00i�ðn�mÞ þ permutationsffiffiffidp ; ð10:29Þ

where the number d of the mutually orthogonal terms also yields the dimension ofthe subspace spanned by themselves. The above expression is just the (7.3)Schmidt representation of the post-measurement state. The coefficients of thed mutually orthogonal tensor product states are all equal to 1=

ffiffiffidp

and this state istherefore maximally entangled (7.11) on the d-dimensional subspace. Suppose logd is integer, it can be shown this is of no ultimate restriction. Then, according toSect. 7.1.6, the state (10.29) is equivalent with

k ¼ log d ¼ nS ð10:30Þ

uncorrelated maximally entangled Bell states jUþABi: This way we have shown thatn partially entangled qubit pairs each of entanglement E can, via LOCC, be dis-tilled into at least k = nE maximally entangled pairs.

10.7 Entanglement Dilution

Suppose Alice and Bob share k examples of the maximally entangled Bell statesjUþABi; let k be large. We show that, using LOCC only, Alice and Bob can dilutetheir states into n* k/E = k/S partially entangled states jWABi of entanglementE = S:

104 10 Q-Information Theory

jUþABi�k �! jWABi�n: ð10:31Þ

To this end, Alice and Bob will combine q-data compression and teleportation.In addition to her k qubits, Alice prepares n = k/S = k/E local examples of the

desired state jWABi of partial entanglement E = S:

cos h j0; A0i � j0; A00i þ sin h j1; A0i � j1; A00ið Þ�n� jWA0A00 i�n: ð10:32Þ

In addition to his k qubits, also Bob prepares n = k/S = k/E local raw qubits inany states, say in a certain q�n

B0 : If Alice can, using LOCC only, transmit eachqubits from her systems A00 to Bob’s systems B0 then Alice and Bob would sharen example of the desired state jWA0B0 i of partial entanglement E = S. Indeed, Alicecan initiate teleportation which is typical LOCC operation. But Alice and Bob can

teleport k qubits only since they share just k Bell pairs jUþABi�k initially. Fortu-

nately, q-data compression will map the n qubits into k.Consider the collective state (10.32) and calculate the reduced collective state

of the n qubits on the systems A00:

cos2 h j0; A00ih0; A00j þ sin2 h j1; A00ih1; A00j� ��n� q�n

A00 : ð10:33Þ

This is the state Alice must teleport to Bob. Observe that qA ¼ qB ¼ qA0 ¼ qA00 ; cf.(10.22), therefore the corresponding von Neumann entropies S coincide. Alicemust compress q�n

A00 : We learned in Sect. 10.3 that for asymptotically large n thecollective state q�n

A00 of n qubits can faithfully be compressed into the 2nS = 2k

dimensional subspace Xn of just k qubits. Therefore Alice teleports these k qubits

(the compressed q-codes) into the subspace Xn of k qubits of Bob’s raw state q�nB0 :

For n ? ?, the resulting state of the non-local composite systems A0B0 becomesthe n = k/S = k/E partially entangled states jWA0B0 i; identical to the desired onesjWABi:

10.8 Entropically Irreversible Operations

We introduced the qualitative notion of q-irreversibility on examples, e.g., ofq-measurements in Sect. 4.4 and reduced dynamics in Sect. 4.5, where it meantjust the non-invertibility of the corresponding q-operations. The informaticalconcept of q-irreversibility is more restrictive than the concept of non-invertibility.It assumes q-information loss or, in alternative terminology: q-entropy increase. Innon-selective q-measurements the von Neumann entropy is always increased orpreserved. In typical reduced dynamics, however, the q-entropy for some initialq-states is increasing while for some other ones it is decreasing. Below, we aregoing to characterize the general class of entropy increasing q-operations.

A non-invertible (non-unitary) q-operationM is (entropically) irreversible if itincreases or preserves the von Neumann entropy:

10.7 Entanglement Dilution 105

SðMqÞ� SðqÞ; ð10:34Þ

for all q-states q. We shall show that a non-invertibleM is irreversible if it leavesthe totally mixed q-state q0 ¼ I=d invariant: Mq0 ¼ q0:

The q-entropy of a state q is always equal to the Shannon entropy computedfrom the eigenvalues {wk}, provided we take the degenerate eigenvalues twice ormore, accordingly. Similarly, the q-entropy of q0 � Mq coincides with theShannon entropy computed from the eigenvalues {w0l} of q0: Therefore it is suf-ficient if we prove the classical relationship S(w0) C S(w) where w, w0 are thediagonal (eigenvalue) distributions of the density matrices q; q0; respectively. Let

us introduce the spectral expansions q ¼P

k wkPk and q0 ¼P

l w0lP0l: Then we

express the eigenvalues of q0 through the following equivalent expressions:

w0l ¼ tr P0lq0

� �¼ tr P

0lMq

� �¼X

k

tr P0lMPk

� �wk; ð10:35Þ

where the last form has been obained by substituting the spectral expansion of q:For fixed w, w0, the above map looks like a classical operation on w, yielding w0

through a transfer function. A transfer function is always normalized in the firstvariable l. Our transfer function is normalized in the second variable as well:

Xk

tr P0lMPk

� �¼ tr P

0lMI

� �¼ 1; ð10:36Þ

since MI ¼ I according to our initial postulation. Our transfer function is doublestochastic, therefore the Shannon entropy of w will be greater or equal to theShannon entropy of w0, cf. Sect. 9.8. In general, the strict inequality holds forq 6¼ q0 unless q is singular and the restriction of the non-unitaryM for the supportof q becomes invertible.

The simplest example of (entropically) irreversible q-operations is the non-selective q-measurement. It is non-invertible and it preserve the totally mixedstate. As a contrary example, consider a ‘‘broken’’ q-channel whose output mes-sage is a permanent j0i for any input whatsoever: q0 ¼ P0qP0 þ rxP1qP1rx wherePx ¼ jxihxj; for x = 0, 1. The q-channel operation is obviously non-invertible butit is not entropically irreversible.

10.9 Problems, Exercises

10.1 Subadditivity of q-entropy. Using the Klein inequality, let us prove thesubadditivity of the von Neumann entropy.

106 10 Q-Information Theory

10.2 Concavity of q-entropy, Holevo entropy. When we prepare a q-state as amixture q ¼

Pn wnqn then the entropy of the mixture is never smaller than

the average entropy of the components:

0� SðqÞ �X

n

wnSðqnÞ:

The r.h.s. is called Holevo entropy. It puts an upper bound on the accessibleinformation and it is bounded from above by the so-called entropy ofpreparation S(w):

A� SðqÞ �X

n

wnSðqnÞ� SðwÞ:

The first inequality is called the Holevo bound [2] and it is a most powerfultheorem of q-information theory.Holevo entropy is non-negative because of the concavity of the q-entropy.Let us prove concavity from subadditivity.

10.3 Data compression of the non-orthogonal code. Suppose random bits areencoded into non-orthogonal pure states exactly like in the case of theq-banknote 6.4.1 as well as of q-cryptography 6.4.2. Thus we consider thestates j"zi; j"xi the q-code of the random classical binary messageX = {x, q(x)}:

x letter: 0 1q-code: j"zi j"xiqðxÞ probability: 1=2 1=2

ð10:37Þ

The Shannon entropy of the random classical letter is H(X) = 1 bit. Let usdetermine the maximum faithful q-data compression rate.

10.4 Distinguishing two non-orthogonal qubits: various aspects. We have previ-ously mentioned two strategies 6.3 to distinguish non-orthogonal randomstates like j"zi; j"xi: The first strategy 6.3.1 used a single projective mea-surement of either rz or rx chosen at random. It was perfectly conclusive atrate 25%. The second strategy 6.3.2 used a single non-projective measure-ment regarding three suitable q-effects Pz; Px; P?; and this strategy wasperfectly conclusive at rate 1� 1=

ffiffiffi2p 30% and perfectly inconclusive in

the rest. In the context of Prob. 10.3, the information gain Igain will coincidewith the rate 30% of the unambiguous decisions. Yet, there are differentprojective q-measurements which are never perfectly conclusive but theirinformation gain is bigger. Let us calculate Igain for the single polarizationmeasurement in the direction (1, 0, -1).

10.5 Simple optimum q-code. Assume we have to encode the three colors R,G, andB, occuring at random, into the respective pure states jRi; jGi; jBi of a qubit.

10.9 Problems, Exercises 107

Suppose we do not care if the q-code would not guarantee the highestaccessible information. Suppose we have limited q-channel capacitiestherefore we want optimum q-code in a sense that it cannot be shortenedfurther, i.e., without the loss of the accessible information. Then we use ourq-channel economically. Let us construct the corresponding three pure statesjRi; jGi; and jBi:

References

1. Schumacher, B.: Phys. Rev. A 51, 2738 (1995)2. Holevo, A.S.: Problems Inf. Transm. 5, 247 (1979)

108 10 Q-Information Theory

Chapter 11Q-Computation

Classical digital computer performs its algorithm via one- and two-bit operationsrealized by one- and two-bit ‘‘gates’’. The (yet hypothetical) q-digital computerwould do the same. The classical logical gates become q-gates, their ‘‘circuits’’ arecalled q-circuits. They can perform what is called q-parallel computation.Q-algorithms are always reversible. Therefore we learn classical reversible com-putation first then we discuss the earliest q-algorithms that might overcome allclassical algorithms targeting a similar task.1

11.1 Parallel Q-Computing

A classical computer works like this: the input storage stores an n-digit binarynumber x upon which the algorithm is performed and the result will be read outfrom the same storage. Now, let the computer be a q-system, operating coherentlyon its q-storage which is an n-qubit state vector

x1j i � . . .� xnj i � xj i; x � x1x2 . . . xn: ð11:1Þ

Unprecedented in classical computation, we can store all possible N = 2n initialvalues in the q-storage if we superpose them, say, with the same amplitudes:

Sj i ¼ 1ffiffiffiffiNp

XN�1

x¼0

xj i: ð11:2Þ

We call this state the totally superposed state of the storage. What about a coherentq-algorithm? It is a unitary N 9 N matrix U: Given a classical algorithm, wewould naively think that its q-counterpart corresponds to a certain matrix U: This

1 For a short review on q-computation, see Ekert et al. [1].

L. Diósi, A Short Course in Quantum Information Theory,Lecture Notes in Physics, 827, DOI: 10.1007/978-3-642-16117-9_11,� Springer-Verlag Berlin Heidelberg 2011

109

is the case, indeed, provided the classical algorithm is reversible. If it is not thenwe have to replace it by a reversible version. Fortunately, it had been known inclassical theory of computation that any computational task can be realizedreversibly if some redundant data are drawn along, cf. Sect. 11.2. This way we canrealize any classical computational task via the corresponding unitary transfor-mation U on a q-storage. Parallel computation becomes possible on all possibleinitial data captured by the totally superposed state Sj i: Moreover, we couldperform various different reversible algorithms running simultaneously on thesame q-computer.

Contrary to early expectations [2], it is not so easy to find a concrete compu-tational task whose coherent q-algorithm outperforms all possible incoherentclassical algorithms. The total superposition of the initial states Sj i is instrumentalbut, surprisingly, it does not in itself guarantee an advantage. We need to inventgenuine q-algorithms, like those of Sects. 11.3 and 11.4, differing radically fromthe classical ones.

11.2 Evaluation of Arithmetic Functions

We are going to discuss a basic classical computational task. The algorithm shouldcompute functions y = f(x) that map n-bit integers x onto m-bit integers y:

ð11:3Þ

The map f is not necessarily invertible, therefore the corresponding algorithm isnot necessarily reversible. For the purpose of q-parallel computation, we extendthe map f in such a way that the new map F be reversible even if f were not.Accordingly, we set up independent input and output storages, i.e., the outputstorage exists already at input time. Let the new invertible map F be the following:

ð11:4Þ

The summation is understood mod 2m: How does this algorithm calculate f(x)?That’s simple. We set the output storage at initial time to zero, y = 0, so that thesame storage at final time will contain f(x).

The new function F(x, y) is always invertible. Therefore the computation of thearithmetic function f(x) has been realized by a reversible classical algorithm whichcan directly be converted into the following q-algorithm:

110 11 Q-Computation

ð11:5Þ

Instead of this diagrammatic definition, the 2n+m 9 2n+m unitary matrix of the q-algorithm can be defined in the usual way as well:

UF xj i � yj ið Þ ¼ xj i � y� f ðxÞj i: ð11:6Þ

If we set the input storage to the totally superposed state jSi instead of jxi and weset the output storage to j0i then the following state emerges:

UF Sj i � 0j ið Þ ¼ 1ffiffiffiffiNp

XN�1

x¼0

xj i � f ðxÞj i: ð11:7Þ

In a single unitary step one has thus evaluated the arithmetic function f (x) for allpossible arguments x = 0, 1, …, N - 1. However, we can read out just one valuewhen we measure the final states of both the input and output registers. Theadvantage of parallel q-computation is related to the usage of the state jSi butrequires more than that.

11.3 Oracle Problem: The First Q-Algorithm

Consider a black-box (11.3) to evaluate a certain arithmetic function f(x), and call itthe (classical) oracle. If we ask the oracle a ‘‘question’’ x, it provides the immediate‘‘answer’’ y = f(x). This time it is irrelevant how it evaluates the function. Supposethe point is that we do not know the function f(x), only the oracle knows it. Thehypothetical task will be this: we want to learn the function f(x) therefore we aregoing to ask questions and evaluating the answers. How many questions shouldwe ask?

It is this class of problems where the first ever task has been invented [3, 4] forwhich a smart q-algorithm is clearly more efficient than any classical algorithm. Inthe simplest case, let f map a single bit x into a single bit y. Suppose we have tolearn whether f(x) is a constant function or it is not. If f(x) is evaluated by aclassical oracle then we must ask twice: we submit both initial values x = 0 andx = 1 in separate ‘‘questions’’. More questions do not exist, less questions are notsufficient to decide whether or not f(0) = f(1). If, however, the function f isevaluated by a q-oracle (11.5) then a single ‘‘question’’ will be sufficient:

11.2 Evaluation of Arithmetic Functions 111

ð11:8Þ

The input storage should be set to the total superposition jSi of all possible states(i.e.: x = 0,1) in accordance with the concept of q-parallel computation (11.2). Wedo not set the output storage to zero, rather we set it to the superposition of the twopossible states x = 0,1—with the opposite phases. We show that to such a single‘‘question’’ the q-oracle’s answer is decisive regarding the constancy of f. As we seefrom (11.8), the output storage never changes. The input storage, however, does:

0j i þ 1j iffiffiffi2p if f ð0Þ ¼ f ð1Þ ;

0j i � 1j iffiffiffi2p if f ð0Þ 6¼ f ð1Þ :

ð11:9Þ

These two input storage states are orthogonal. A single ultimate projective q-measurement can discriminate them. After the measurement we know with cer-tainty whether the function f is constant or not.

We have yet to prove that the q-oracle provides the ‘‘answer’’ (11.8) for thesmart ‘‘question’’ there. Obviously we can re-write the initial state this way:

0j i þ 1j iffiffiffi2p � 0j i � 1j iffiffiffi

2p ¼ 1ffiffiffi

2p

Xx¼0;1

xj i � 0j i � 1j iffiffiffi2p : ð11:10Þ

The unitary map UF (11.6) applies to this, and yields the following state:

1ffiffiffi2p

Xx¼0;1

xj i � f ðxÞj i � 1� f ðxÞj iffiffiffi2p : ð11:11Þ

We can write the state of the output storage into an equivalent form

f ðxÞj i � 1� f ðxÞj i ¼ ð�1Þf ðxÞ 0j i � 1j ið Þ : ð11:12Þ

Indeed, the state of the output storage has, upto its phase, remained unchanged.Using the above form, the final composite state produced by the q-oracle becomes

1ffiffiffi2p

Xx¼0;1

ð�1Þf ðxÞ xj i � 0j i � 1j iffiffiffi2p : ð11:13Þ

It coincides with the ‘‘answer’’ displayed on the diagram (11.8).The advantage of q-tests over classical ones becomes more spectacular when

the oracle f maps a large number of n bits into one. Suppose the arithmetic functionf is either constant or balanced. This latter means that f(x) returns 0 and 1 equal

112 11 Q-Computation

times when x runs over the N = 2n different values. Suppose that we are interestedto learn whether the oracle is constant or balanced. Classically we must askN/2 ? 1 ‘‘questions’’. In case of the corresponding q-oracle, a single ‘‘question’’ isenough. The construction is very similar to the former simplest one. We keepsetting the input storage to the totally superposed state jSi and we keep setting theoutput storage to ð 0j i � 1j iÞ=

ffiffiffi2p

: We analyze the ‘‘answer’’ of the oracle by anultimate projective q-measurement of jSi hSj. We are going to prove that f isconstant if the outcome is 1 and f is balanced if the outcome is 0. The proofconsists of the same steps as before, with the only change that x runs from 0 toN - 1. Accordingly, the initial state reads

Sj i � 0j i � 1j iffiffiffi2p ¼ 1ffiffiffiffi

Np

XN�1

x¼0

xj i � 0j i � 1j iffiffiffi2p ; ð11:14Þ

while the final state, i.e., the ‘‘answer’’ will be

1ffiffiffiffiNp

XN�1

x¼0

ð�1Þf ðxÞ xj i � 0j i � 1j iffiffiffi2p : ð11:15Þ

Obviously, the final state of the input storage is jSi when f is constant and it isorthogonal to jSi when f is balanced.

11.4 Searching Q-Algorithm

Suppose that the function f(x), mapping n bits into one bit, is always zero exceptfor x = x0:

f ðxÞ ¼ dxx0: ð11:16Þ

Let a classical oracle evaluate the function. Assume that x0 is unknown to us andwe have to learn it by asking the oracle. This is the basic searching problemformulated in the language of the oracle-problem. The unknown integer x0 cantake any values between 0 and N - 1 where N = 2n. Obviously, we must askN - 1 questions in the worst case, and O(N) questions in general . Grover showedthat we can ask Oð

ffiffiffiffiNpÞ questions of the corresponding q-oracle, which is much

less than O(N). The corresponding q-algorithm [5] will be much faster than anyclassical algorithm to solve the same searching problem.

Grover’s q-algorithm is iterative, of R�ffiffiffiffiNp

iterations. In each cycle, we askthe q-oracle one question and perform a given unitary transformation on itsanswer. This state becomes the question in the forthcoming cycle, etc. The outputstorage of the oracle stays always in the state ð 0j i � 1j iÞ=

ffiffiffi2p

; hence the ‘‘ques-tion’’ jxi submitted into the input storage will produce this answer:

ð�1Þ f ðxÞ xj i ¼ ð1� 2dxx0Þ xj i: ð11:17Þ

11.3 Oracle Problem: The First Q-Algorithm 113

This ‘‘question-to-answer’’ map can be described by the following N 9 N unitarymatrix:

I � 2 x0j i x0h j: ð11:18Þ

Thus we use the q-oracle to mark for us the searched direction jx0i in the state-space. Let our first ‘‘question’’ be the totally superposed state jSi. The ‘‘answer’’will be ðI � 2 x0ihx0j jÞ jSi and, according to Grover, we transform it by anotherunitary matrix

I � 2 Sj i Sh j: ð11:19Þ

The result will be utilized as the ‘‘question’’ asked of the q-oracle in the seconditeration cycle, etc. We are going to prove that after R iterations the resulting stateis approximately jx0i:

ðI � 2 Sj i Sh jÞðI � 2 x0j i x0h jÞ� �R

Sj i � x0j i ; ð11:20Þ

so that the projective q-measurement of x on the resulting state yields the searchedvalue x0.

The proof is elementary. Why? The iteration (11.20) does nothing else than arotation of the initial vector jSi toward the searched vector jx0i, by the same anglein each cycle. The process can be described in the real two-dimensional planespanned by jSi and jx0i . In this plane, rotations (11.18, 11.19) are consideredmodulo p since the sign of the state vector is irrelevant. For simplicity’s sake, weassume N � 1. Then jx0i is almost orthogonal to jSi :

x0jSh i ¼ 1ffiffiffiffiNp : ð11:21Þ

Their angle ðp=2Þ � � is almost the rectangle. The defect is very small:

� � 1ffiffiffiffiNp : ð11:22Þ

In the first iteration, the matrix (11.18) reflects the initial state jSi w.r.t. thedirection jx0i; the resulting vector will be reflected by the matrix (11.19) w.r.t. thedirection of jSi. The outcome vector is such that it encloses a smaller angleðp=2Þ � 3� with the searched vector jx0i . It is easy to see that the two reflectionsin each further cycle of Grover iteration will always rotate toward jx0i by the sameangle 2�. Since the initial vector jSi was away from jx0i by an angle ðp=2Þ � �, thenecessary number of iterations will be

R � p4�� p

4

ffiffiffiffiNp

: ð11:23Þ

114 11 Q-Computation

The R-times iterated state approaches jx0i to within a small angle � �. Hence itsprojective measurement in the computational basis yields x0 with an error prob-ability as little as * 1/N.

11.5 Fourier Algorithm

The well-known discrete classical Fourier transformation is equivalent with aunitary transformation on the given complex vector space. We can use theq-physics notations. Let {jxi} stand for the basis vectors where x runs through theN binary numbers of n = log N bits. The classical Fourier transformation is givenby

U jxi ¼ 1ffiffiffiffiNp

XN�1

y¼0

e2pixy=N yi:j ð11:24Þ

Equivalently, the elements of the N 9 N unitary Fourier matrix read

yh jU xj i ¼ 1ffiffiffiffiNp e2pixy=N : ð11:25Þ

Since this classical map is reversible we can directly turn it into a q-algorithm. Yet,as we shall see, the entanglement of U xj i is only apparent, it is a multiple directproduct state in fact. This factorization makes the classical fast-Fourier algorithmpossible.

We are going to prove that, using the notation x = x1x2 … xn, the Fouriertransform is a product state of n different qubits. We substitute the digitalexpansion y ¼ N

Pnk¼1 yk=2k in order to factorize the elements of the unitary

matrix of the Fourier transform:

e2pixy=N ¼Yn

k¼1

expð2pixyk=2kÞ ¼ e2piy10:xn e2piy20:xn�1xn . . . e2piyn0:x1x2 ... xn : ð11:26Þ

We can also factorize the sum over the computational basis states:

Xy

e2pi... yj i ¼X1

y1 ¼ 0

X1

y2 ¼ 0

. . .X1

yn ¼ 0

e2pi... y1j i � y2j i � � ynj i: ð11:27Þ

Substituting these expressions into that of the Fourier transform (11.24), we obtainthe following:

U xj i ¼ 0j i þ e2piy10:xn 1j iffiffiffi2p � 0j i þ e2piy20:xn�1xn 1j iffiffiffi

2p � . . .� 0j i þ e2piyn0:x1x2 ... xn 1j iffiffiffi

2p

ð11:28Þ

11.4 Searching Q-Algorithm 115

This shows that the q-algorithm of the Fourier transform can be built up fromsingle qubit operations controlled by the state of other qubits. Thus, in fact, theseoperations are multi-qubit operations. Although the number of control qubits ismore than one, two-qubit controlled gates (cf. Sect. 11.8) are universally sufficientwhen organized in suitable q-circuits, cf. Fig. 11.2.

11.6 Period Finding Q-Algorithm

Let f(x) be an arithmetic function that maps n = logN bit integers into m bitintegers. Suppose it is periodic with period r, i.e., for all 0 B x B N - r - 1:

f ðxþ rÞ ¼ f ðxÞ : ð11:29Þ

Suppose that r is unknown to us and we have to determine it. According to thewidespread classical method, we calculate the Fourier transform of f(x):

~f ðyÞ ¼ 1ffiffiffiffiNp

XN�1

x¼0

e2pixy=Nf ðxÞ ; ð11:30Þ

which will show peaks at the multiples of the wave number N/r. The proof is thefollowing. We consider the general case when r does not divide N while weassume that the integer number m = [N/r] of complete periods is sufficiently big. If

we approximatePN�1

x¼0 byPmr�1

x¼0 we obtain

~f ðyÞ�Xmr�1

x¼0

e2pixy=Nf ðxÞ ¼Xm�1

‘¼0

e2pi‘ry=N

!Xr�1

x¼0

e2pixy=Nf ðxÞ: ð11:31Þ

The pre-factor takes the closed form

Xm�1

‘¼0

e2pi‘ry=N ¼ 1� e2pimry=N

1� e2piry=N; ð11:32Þ

which has peaks if y is close to a multiple of N/r.Similar is the Fourier q-algorithm of period finding. We evaluate the function

f(x) for all arguments x = 0,1, … ,N - 1 in a single unitary operation (11.7), i.e.,we prepare the following state:

1ffiffiffiffiNp

XN�1

x¼0

xj i � f ðxÞj i ; ð11:33Þ

but we do not yet read it out. Rather we apply the Fourier transformation (11.24) tothe input register, yielding

116 11 Q-Computation

1ffiffiffiffiNp

XN�1

x¼0

U xj i � f ðxÞj i ¼ 1N

XN�1

y¼0

yj i �XN�1

x¼0

e2pixy=N f ðxÞj i: ð11:34Þ

We measure the computational basis in the input state. The random measurementoutcome y will have the following probability:

pðyÞ ¼ 1N

XN�1

x¼0

e2pixy=N f ðxÞj i�����

�����2

: ð11:35Þ

This probability distribution is peaked if y is close to a multiple of N/r. The proofcopies the classical proof: the approximation and reorganization of the sum in(11.30) can invariably be applied to the sum on the r.h.s. above. Finally we get the(squared modulus of the) classical pre-factor (11.32) of ~f ðyÞ as the pre-factor ofp(y):

pðyÞ� 1� e2pimry=N

1� e2piry=N

��������2

Xr�1

x¼0

e2pixy=N f ðxÞj i�����

�����2

: ð11:36Þ

This probability is enhanced at the multiples of N/r. Hence the outcome y is verylikely to be the multiple of N/r, i.e., y/N& k/r where both integers k and r areunknown. To find r we look for the best rational number approximation toy/N where the divisor is less (in our case: much less) than N, and we identify r bythat divisor. If a check (11.29) fails, we repeat the whole algorithm.

The period finding q-algorithm becomes instrumental for the Shor algorithm [6]to factorize large numbers, which is the most promising prediction of q-compu-tation theory. The Shor q-algorithm is exponentially faster than all known classicalalgorithms to factorize large integers. This speed-up of the Shor algorithm is due tothe q-algorithm of period finding as part of the Shor algorithm.

11.7 Error Correction

Uncontrollable environmental interactions, i.e., noise may corrupt the storage ofcomputers. Error detection and correction of a classical storage are possible if weapply redundant codes. If we encode one logical bit into three identical raw bitsthen all bit-flip errors can be detected and corrected by ‘‘majority voting’’ providedthe noise corrupts a single raw bit at a time. The corrupted q-storage is a little moredifficult task because error detection and correction must respect and restore thefaultless superpositions and entanglements. We must detect and correct the error ofa qubit whereas the qubit itself remains unknown all the time.

Consider the error detection and correction of a single qubit jxi: First wediscuss how to protect the qubit c0 j0i þ c1 j1i against a bit-flip error. The sim-plest idea is that we encode our logical qubit into three raw qubits:

11.6 Period Finding Q-Algorithm 117

wj i ¼ c0 000j i þ c1 111j i; ð11:37Þ

which occupy the two-dimensional code space

C ¼ 000j i 000h j þ 111j i 111h j ð11:38Þ

within the eight-dimensional raw space. Suppose the first qubit gets corrupted by abit-flip:

wj i ! w0j i ¼ X1 wj i ¼ c0 100j i þ c1 011j i : ð11:39Þ

Observe that the unitary bit-flip maps the code space C onto the subspaceP1 ¼ X1CX1 which is orthogonal to C: Alternatively, if the bit-flip error corruptedthe second or third qubit, the code space would map to P2 ¼ X2CX2 or P3 ¼X3CX3; respectively. The projectors P1; P2; P3—called error syndromes—form anorthogonal set. If we measure them on the state jw0i, we can detect the bit-fliperror, and we can correct it. If the k0th qubit (k = 1, 2, 3) has been corrupted thenthe measurement outcome is Pk ¼ 1; and we know from it that we have to re-flipthe k0th qubit:

w0j i ! Xk w0j i ¼ wj i: ð11:40Þ

The unknown original state (11.37) recovers provided the bit-flip error has influ-enced just one of the three raw qubits.

A similar protocol applies if, instead of the bit-flip errors X1; X2; X3; we assumeanother unitary equivalent set of single-qubit errors like, e.g., the phase-flip errorsZ1; Z2; Z3: Since Zk ¼ HkXkHk for k = 1, 2, 3, we must apply the unitary trans-form H1H2H3 to the basis vectors fj000i; j111ig as well as to the error syndromesPk of the bit-flip error correction protocol, this way we obtain the phase-flip errorcorrection protocol.

The real issue is whether we can construct error correction protocols for allsingle qubit errors X; Y; Z together. Three raw qubits are not sufficient because theeight-dimensional Hilbert space can not host more than three error syndromes. Butnine redundant raw qubits are already sufficient [7]:

wj i ¼ c0000j i þ 111j iffiffiffi

2p

� ��3

þc1000j i � 111j iffiffiffi

2p

� ��3

: ð11:41Þ

Then the code space reads

C ¼ 000j i þ 111j iffiffiffi2p 000h j þ 111h jffiffiffi

2p

� ��3

þ 000j i � 111j iffiffiffi2p 000h j � 111h jffiffiffi

2p

� ��3

:

ð11:42Þ

There can be 9 9 3 = 27 different single-qubit errors:

118 11 Q-Computation

wj i ! w0j i ¼ ðraÞk wj i ða ¼ x; y; z; k ¼ 1; 2; . . .; 9Þ ð11:43Þ

where, for notational convenience, we returned to Pauli’s symbols instead ofX; Y ; Z: The error syndromes corresponding to the 27 errors are Pak ¼ ðraÞkCðraÞk:One can directly inspect that these 27 projectors and C are all mutually orthogonal,the proof needs just patience. Now we possess an error correcting protocol wherenine raw qubits encode one logical qubit, and we can correct any of the three unitaryerrors rx; ry; rz corrupting any single qubit. We measure the 27 projectors, if Pak ¼1 for a given pair a, k then we know that the k0th qubit has been corrupted by theunitary operation ra: Let us apply the same unitary transformation ra once more tothe k0th qubit of the state jw0i, it recovers the correct state jwi (11.41).

However, the environmental influence on a single qubit can be much morecomplex than (11.43). It is, in general, the statistical mixture of linear maps (cf.Sect. 8.2), the resulting (unnormalized) state of a linear map can be written as

wj i ! w0j i ¼ Ik þ vxðrxÞk þ vyðryÞk þ vzðrzÞk

wj i; ð11:44Þ

where vx, vy, vz are complex numbers. There is a continuum of different errors!The ultimate good news is this: the nine-qubit protocol with its 27 different errorsyndromes will correct all single qubit errors. Indeed, the measurement of the 27error syndromes first collapses the corrupted state jw0i into one of its four termsabove. No correction is needed if all syndromes yield zero. In the contrary case,the discrete error correction recovers the same correct state jwi (11.41).

11.8 Q-Gates, Q-Circuits

Any given classical algorithm can be decomposed into a series of Boolean operations.It is known that a small selection of one- and two-bit operations suffice although thisselection is not unique. The selected ones are called universal logical operations (orgates, referring to their technical realization). To perform a given algorithm, theclassical logical gates are organized into logical circuits. Similarly, we defineq-logical gates and circuits to perform q-algorithms. We have already learned aboutthe 1-qubit logical operations (Sect. 6.1.1) and now we associate the notion of q-gateto them. We also mentioned that the Hadamard gate (H) and phase-gate (T), i.e.:

ð11:45Þ

ð11:46Þ

11.7 Error Correction 119

are universal one-qubit gates: all one-qubit unitary operations can be realized bythem.

The two-qubit operations (gates) can be derived from the so-called controlledoperations (gates). Then the first qubit is the control-qubit and the second is thetarget-qubit. If the control-qubit is 1 then a given unitary operation U is performedon the target-qubit. If the control qubit is 0 then nothing will happen:

ð11:47Þ

Fig. 11.1 Q-circuit encodingShor error correction. Thisstructure (cf. [9] by Nielsenand Chuang), can easily becompared with the desiredcode state (11.41)

Fig. 11.2 Q-circuit of the Fourier transform. This three-qubit (n = 3) example illustrates thegeneral structure, cf. [9] by Nielsen and Chuang. An overall phase, not shown here, w.r.t. theexpression (11.28) comes from the conventional choice (11.46) of the phase-gates

120 11 Q-Computation

This is, for instance, the controlled NOT (cNOT, XOR):

ð11:48Þ

The 4 9 4 matrix corresponds to the order j00i; j01i; j10i; j11i in obviousnotation jxyi ¼ jxi � jyi of the bipartite basis-vectors. The chosen symbol � onthe diagram reminds us that in computational basis the cNOT sets the target-qubitto the modulo 2 sum of the two initial qubits.

It is known that the Hadamard-, the phase-, and the cNOT-gates are universalq-gates, cf. Problem 6.1. These three q-gates allow constructing all unitary oper-ations on a q-storage of any number n of qubits. Design of q-logical circuits,similarly to design of classical logical circuits, has its own standards and tricks [8].Talking about q-algorithms, standard diagrams of q-circuits are used extensively.Moreover, the language and formalism of q-information theory is penetrating ourunderstanding—and teaching—the traditional q-theory as well.

In order to illustrate the art of q-circuit design, we include two examples whichgive a first hint and further motivations to the interested.

11.9 Problems, Exercises

11.1 Creating the totally symmetric state. Let us prove that n independent qubits,in state j0i each, can be transformed into the totally symmetric state jSi byusing n Hadamard gates.

11.2 Constructing Z-gate from X-gate. Let us confirm that the one-qubit q-gatesX and Z are related via Hadamard gate sandwiches: HXH = Z and HZH = X.

11.3 Constructing controlled Z-gate from cNOT-gate. Adding one-qubit gates to acNOT-gate in a suitable way, let us build a q-circuit to realize a controlled-Zoperation.

11.4 Constructing controlled phase-gate from two cNOT-gates. Adding one-qubitgates to two cNOT-gates in a suitable way, let us build a q-circuit to realize acontrolled-phase operation.

11.5 Q-circuit to produce Bell states. Let us show that the following q-circuitproduces the four Bell states from the computational basis:

ðÞ

11.6 Q-circuit to measure Bell states. Let us construct the q-circuit of projectivemeasurement in the Bell basis.

11.8 Q-Gates, Q-Circuits 121

References

1. Ekert, A., Hayden, P., Inamori, H.: Basic concepts in quantum computation (Les Houcheslectures 2000); Los Alamos e-print arXiv: quant-ph/0011013

2. Feynman, R.P.: Int. J. Theor. Phys. 21, 467 (1982)3. Deutsch, D.: Proc. R. Soc. Lond. A 400, 97 (1985)4. Deutsch, D., Jozsa, R.: Proc. R. Soc. Lond. A 439, 533 (1992)5. Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of the

28th Annual STOC, Association for Computer Machinery, New York (1996)6. Shor, P.W.: Algorithms for quantum computation: discrete logarithm and factoring. In: 35th

Annual Symposium on Foundations of Computer Science. IEEE Press, Los Alamitos (1994)7. Shor, P.W.: Phys. Rev. A 52, 2493 (1995)8. Tucci, R.R.: QC Paulinesia. http://www.ar-tiste.com/PaulinesiaVer1.pdf (2004)9. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge

University Press, Cambridge (2000)

122 11 Q-Computation

Chapter 12Qubit Thermodynamics

Entropy had been rooted in thermodynamics. We have to explore this backgroundof the microscopic q-entropy. We do not hesitate to apply thermodynamics to asingle qubit if we recall the classical ideal gas where thermodynamical equationsare satisfied by the microscopic state of each single molecule. Since thermalenvironment is natural for a physical qubit, further notions of thermodynamics,like thermalization, refrigeration, or the Carnot cycle, apply to qubits. Their sig-nificance for q-information is a delicate challenge.

12.1 Thermal Qubit

According to the general principles of thermodynamics, a system placed into athermal reservoir of temperature T will reach thermal equilibrium with the res-ervoir. This principle is valid for a quantum system as well: it will under theinfluence of the reservoir relax to a well-defined thermal equilibrium state, the so-called Gibbs state N expð�H=kBTÞ (kB is the Boltzmann constant) which isobviously a stationary state of the von Neumann equation (4.5). When the quantumsystem is a single qubit, we can write the Hamiltonian matrix (5.24) in the formH ¼ � 1

2 �rz: This may correspond to an electronic spin in vertical magnetic fieldwhere � is the energy difference between the states j"i ¼ j0i and j#i ¼ j1i: Thesame Hamiltonian matrix may also refer to a two-level atom where we identify itsground and excited states as j0i and j1i; respectively. The Gibbs state of the qubittakes the form

qb ¼1

2 coshðb�=2Þeb�rz=2 ¼ 1

1þ expð�b�Þ j0ih0j þ e�b�j1ih1j� �

: ð12:1Þ

L. Diósi, A Short Course in Quantum Information Theory,Lecture Notes in Physics, 827, DOI: 10.1007/978-3-642-16117-9_12,� Springer-Verlag Berlin Heidelberg 2011

123

We introduced the inverse temperature b ¼ 1=kBT : Recall the Fock representationfrom Sect. 5.5 where we talk about the occupation number n ¼ j1ih1j ¼ aya of theexcited state which in qb yields

nb ¼ trðnqbÞ ¼1

1þ expðb�Þ: ð12:2Þ

For thermodynamic purposes, it is more convenient if we shift the Hamiltonian H

by a constant so that H ¼ �n; i.e., the ground state energy is zero. Let us determinethe average energy of our thermal qubit:

E ¼ trðHqbÞ ¼�

1þ eb�; 0�E� 1

2�: ð12:3Þ

This we shall identify as the thermodynamic energy of the thermal qubit. We alsocalculate the von Neumann entropy of the Gibbs state:

SðqbÞ ¼ � trðqb log qbÞ ¼log 1þ eb�� �1þ eb�

þlog 1þ e�b�� �1þ e�b�

: ð12:4Þ

With one eye on thermodynamics, we express the r.h.s. in terms of the energy E:

SðEÞ ¼ ��� E

�log

�� E

�� E

�log

E

�; 0�E� 1

2; epsilon: ð12:5Þ

This entropy SðEÞ is our candidate thermodynamic function for the thermal qubit.We must give it the right physical dimension by a factor kB; and we have to rescaleit by a factor ln 2 to turn the binary logarithm into the natural one:

SthðEÞ ¼ ðkB ln 2ÞSðEÞ ¼ �kB�� E

�ln�� E

�� kB

E

�ln

E

�: ð12:6Þ

The entropy of a single thermal qubit satisfies the thermodynamics equation

dSthðEÞdE

¼ 1T: ð12:7Þ

12.2 Ideal Qubit Gas

If we want to construct a bulk macroscopic thermodynamic system yet with theexactly tractable quantum dynamics, our simplest system is the abstract ideal gasof qubits. Suppose the gas consists of N non-interacting distinguishable qubits asmolecules, each having the same Hamiltonian H ¼ �n: The thermal equilibriumstate of the gas is

qb � qb � � � � � qb ¼ q�Nb ; ð12:8Þ

124 12 Qubit Thermodynamics

where qb is given by the Eq. (12.1). The average energy E and the von Neumannentropy S of the gas become N times the single qubit average energy (12.3) andentropy (12.5), respectively. Therefore we obtain the thermodynamic entropy ofthe gas if we take N-times the single qubit entropy (12.6) whereas we replace thesingle qubit thermodynamic energy by E=N:

SthðE; NÞ ¼ NðkB ln 2ÞSðE=NÞ: ð12:9Þ

This entropy is an extensive thermodynamic variable itself since it is a homoge-neous linear function of the two extensive thermodynamic variables E and N; as itshould be. The entropy of the ideal qubit gas by construction satisfies the samethermodynamics equation (12.7) as its molecules do:

oSthðE;NÞoE

¼ 1T: ð12:10Þ

Another standard relationship oSthðE;NÞ=oN ¼ �l=T defines the chemicalpotential l of the qubit ideal gas.

The qubit ideal gas has exact additive thermodynamics in the number N of themolecules. This is why single qubit thermodynamics ðN ¼ 1Þ introduced in thepreceding section makes sense though it is hardly conform with the macroscopicnotion of a thermodynamic system which would even suppose the thermodynamiclimit N !1:

12.3 Informatic and Thermodynamic Entropies

For a single qubit as well as for the qubit ideal gas, we derived the simplerelationship between the thermodynamic and informatic (von Neumann) entropies:

Sth ¼ ðkB ln 2ÞS: ð12:11Þ

It turns out that this relationship is valid for general q-systems as well. However,we should ask how it is possible? The interpretation of the informatic entropy Srefers to the maximum compressibility of q-data, it tells us the length of theshortest faithful code that we can reach to compress our data, cf. Chaps. 9 and 10.Apparently, S constrains our success to economize our resources like storage orchannel capacities. On the other hand, the thermodynamic entropy Sth constrainsthe objective physics of the given system. What has Sth to do with S then?Obviously nothing, unless ‘‘data compression’’ is present in the physical worldindependently of human intentions. In fact, this is the case.

We know from statistical physics that the thermal equilibrium of a systemcorresponds to a q-state q which is maximally randomized at the given conditions.By maximally randomized, statistical physics understands the maximum of the

12.2 Ideal Qubit Gas 125

von Neumann entropy. One can directly prove that the Gibbs state qb ¼N expð�bHÞ of a q-system corresponds to the maximum of the von Neumannentropy SðqÞ at fixed average energy E ¼ trðHqÞ: This fact is independent of theinformatic context of the von Neumann entropy. But we can turn it around andgive it the informatic interpretation. The equilibrium q-state qb looks like ashortest code at the constraint of fixed average energy. This shows us that maxi-mum ‘‘data compression’’ is a physical phenomenon. What was the redundantcode, or what was the data? Such relevant further issues, ignored this time, mightlead us to deeper insight into the information aspect of statistical physics and viceversa.

The equivalence (12.11) of thermodynamic and informatic entropies is estab-lished for all systems in thermal equilibrium. A rigorous proof is possible forhomogeneous systems in the limit of infinite size. Whether the equivalence can beextended for non-equilibrium states is a principal issue. The concrete proof is,however, problematic because the full microscopic description of non-equilibriumstates is technically difficult if not completely hopeless.

12.4 Q-Thermalization

As we said in Sect. 12.1, a certain qubit placed into a thermal reservoir of tem-perature T will reach the thermal equilibrium state qb (12.1). We are going toconstruct a simple model1 of this thermalization, and we continue to prefer theFock representation, cf. Sect. 5.5.

Our thermal reservoir will be the abstract qubit gas (Sect. 12.2) in thermalequilibrium. We consider a further ‘‘central’’ qubit in arbitrary initial state q: Thesame Hamiltonian H ¼ �n ¼ �aya will be assumed for the reservoir qubits as wellas for the central qubit. The central qubit is the one to be thermalized, i.e., led tothe state qb by the reservoir. The thermalization will be modelled through inde-pendent two-qubit interactions (collisions) between a reservoir qubit qb and thecentral qubit. Let the interaction correspond to an instantaneous unitary transfor-mation U which occurs at frequency m: If the reservoir is large, it is plausible toassume that no reservoir qubit will interact twice or more, i.e., the central qubitwill always interact with an unperturbed reservoir qubit qb:

Consider a single collision qb � q! Uqb � qUy between a certain reservoir qubit

and the central qubit, described by a certain unitary 4� 4 matrix U: The resultingirreversible q-operation on the reduced density matrix of the central qubit reads

q �! trb Uqb � qUy� �

; ð12:12Þ

1 After [1] by Scarani et al.

126 12 Qubit Thermodynamics

where we refer to the partial trace over the reservoir qubit. Our choice for theunitary matrix U is this:

U ¼ exp gay � a� h:c:� �

; ð12:13Þ

which is the general form to exchange the excitation energy between the reservoirand the central qubits, respectively. If we substitute U into (12.12), we can easilycalculate the r.h.s. to the second order in the coupling g:

q �! qþ g2ð1� nbÞ aqay � 12faya; qg

� �þ g2nb ayqa� 1

2faay; qg

� �: ð12:14Þ

We used the identities trðayaqbÞ ¼ nb and trðaayqbÞ ¼ 1� nb; cf. Sect. 12.1. Notethe appearance of a Lindblad structure (Sect. 8.6). The expression of dq=dt con-sists of the Hamiltonian part and of the above perturbative contribution of thecollisions repeated at frequency m:

dqdt¼ � i

�h½H; q� þ mg2ð1� nbÞ aqay � 1

2faya; qg

� �þ mg2nb ayqa� 1

2faay; qg

� �:

ð12:15Þ

This perturbative result becomes exact in the limit g! 0;m!1 at mg2 ¼const: Let us cast it into the standard form, introduce the constant C ¼ mg2ð1� nbÞand consider that nb ¼ e�b�ð1� nbÞ: In such a way we obtain the form (8.25) ofthe thermalization master equation:

dqdt� Lq ¼ � i

�h½H; q� þ C aqay � 1

2faya; qg

� �þ e�b�C ayqa� 1

2faay; qg

� �:

ð12:16Þ

The second term corresponds to spontaneous decay j1i ! j0i at rate C: The thirdterm corresponds to thermal excitation j0i ! j1i at rate suppressed by the usualBoltzmann factor. The competition between the decay and excitation (emissionand absorption) leads to the state qb whose stationarity can be confirmed bysubstitution into the master equation above.

12.5 Q-Refrigerator

We are going to present a refrigerator2 which operates on two qubits only, incontact with two respective thermal reservoirs—a cold one at temperature Tc and ahot one at temperature Th [ Tc: The refrigerator will develop a colder part than thecold reservoir, able to cool further qubits to a certain T0 below Tc: We know from

2 After [2] by Linden et al.

12.4 Q-Thermalization 127

thermodynamics that it is indeed possible to build refrigerators with no movingparts and with no external resources of energy other than the heat flow from a hotreservoir to a cold one. The peculiarity of our q-refrigerator is that it can operateon a single two-qubit system.

Suppose the first qubit of excitation energy �c is in contact with the coldreservoir, the second qubit of a smaller excitation energy �h\�c is in contact withthe hot reservoir. Both qubits will reach their thermal equilibrium states (12.1) ofinverse temperatures bc ¼ ð1=kBTcÞ and bh ¼ ð1=kBThÞ; respectively:

qc ¼j0; cih0; cj þ expð�bc�cÞj1; cih1; cj

1þ expð�bc�cÞ;

qh ¼j0; hih0; hj þ expð�bh�hÞj1; hih1; hj

1þ expð�bh�hÞ:

ð12:17Þ

The refrigerator itself consists of these two qubits forming a four-state q-system. Theenergy spectrum is 0\�h\�c\�h þ �c: The corresponding four eigenstates are

j0i � j0; ci � j0; hi;j�hi � j0; ci � j1; hi;j�ci � j1; ci � j0; hi;

j�h þ �ci � j1; ci � j1; hi:

ð12:18Þ

The composite state of the refrigerator is qh � qc which yields

j0ih0jþexpð�bh�hÞj�hih�hjþexpð�bc�cÞj�cih�cjþexpð�bh�h�bc�cÞj�hþ�cih�hþ�cj1þ expð�bh�hÞ þ expð�bc�cÞ þ expð�bh�h � bc�cÞ

:

ð12:19Þ

We concentrate our attention on the two-dimensional subspace fj�hi; j�cig whoseexcitation energy is �0 ¼ �c � �h: We can see that the population of the ‘‘excited’’state j�ci is suppressed by a factor expð�bc�c þ bh�hÞ w.r.t. the ‘‘ground’’ statej�hi: This suppression would correspond to thermal equilibrium at a certain tem-perature T0 which we can easily determine if we identify the above factor ofsuppression by the Boltzmann factor expð�b0�0Þ with notation b0 ¼ ð1=kBT0Þ:This way we obtain the constraint bc�c � bh�h ¼ b0�0: It is trivial to solve it for T0:

T0 ¼1� ð�h=�cÞ

1� ðTc=ThÞð�h=�cÞTc; ð12:20Þ

which is smaller than Tc since we supposed that Tc\Th and �c [ �h:This means that the subspace fj�hi; j�cig is colder than the cold reservoir, the

refrigerator can cool a thermometer, i.e., an additional qubit to temperature T0

below Tc: The qubit to be cooled must have the same excitation energy �0 whichthe transition j�hi ! j�ci has, its coupling to the subspace fj�hi; j�cig must pre-serve the energy, and must be weak w.r.t. the thermalization rates of both the hotand cold reservoirs, respectively.

128 12 Qubit Thermodynamics

12.6 Thermal Qubit with External Work

In addition to heat exchange, thermodynamics treats energy exchange in the formof work as well. This concept applies to qubit-thermodynamics whenever we canexternally control the Hamiltonian of the qubit. This happens most typically for theHamiltonian H ¼ �1

2�hxr (5.24) of an electronic or nuclear spin in externalmagnetic field x: We restrict our discussion for the special case where the exci-tation energy � of the qubit can be deliberately controlled by the external verticalmagnetic field xz ¼ x 0:

� ¼ �hx: ð12:21Þ

Our previous results on the thermal qubit remain valid with one exception. We donot shift the ground state energy to zero, rather we retain the eigenvalues 1

2� ¼1

2�hx for the excited/ground states because we are interested in their variationwith the magnetic field.

Suppose we prepare an equilibrium state (12.1) at a certain (inverse) temper-ature b and magnetic field x:

qb;x ¼1

2 coshðb�hx=2Þeb�hxrz=2: ð12:22Þ

The average energy is the expression (12.3) minus 12� ¼ 1

2�hx:

E ¼ trðHqb;xÞ ¼ ��hx2

sinhðb�hx=2Þ; �12�hx� 0: ð12:23Þ

What happens if we change the magnetic field with time and the Hamiltonianbecomes time dependent: HðtÞ ¼ �1

2�hxðtÞrz? The change of the average energy hastwo contributions: the work W exerted on the qubit by the variation of the magneticfield, and the heat Q flowing from the reservoir into the qubit. Accordingly:

dE

dt¼ tr

dH

dtq

� �þ tr H

dqdt

� �� dW

dtþ dQ

dt; ð12:24Þ

where dW=dt is the power and dQ=dt is the rate of heat flow.3 They depend on thedynamics of the qubit, which has two special cases of interest: the isothermal andthe isentropic (or adiabatic4) processes.

In the isothermal process we leave the qubit in contact with the thermal res-ervoir and keep dx=dt much smaller than the thermalization rate C (cf. Sect. 12.4).Hence the qubit remains always thermalized at constant temperature for different

3 Cf. [3] by Alicki, [4] by Spohn and Lebowitz.4 Thermodynamic adiabaticity, meaning ‘‘no heat transfer’’, differs from dynamical adiabaticity,a synonym of being quasi-static, which, e.g., in Chap. 3 stands for ‘‘no excitation’’.

12.6 Thermal Qubit with External Work 129

magnetic fields: qðtÞ ¼ qb;xðtÞ will be a good approximation. The power and theheat flow rate take the following forms, respectively:

dW

dt¼ tr

dH

dtq

� �¼ ��h

2dxdt

sinhðb�hx=2Þ

dQ

dt¼ tr H

dqdt

� �¼ �b�hx

4�h

dxdt

coshðb�hx=2Þ:ð12:25Þ

In the isentropic process we isolate the qubit from the thermal reservoir. Withoutheat transfer, the slow variation of x leaves the process reversible. In our special caseqðtÞ remains the initial equilibrium state qb;x because ½HðtÞ; qb;x� ¼ 0 for all t: Theconstancy of q makes the above defined heat flow vanish: dQ=dt ¼ 0: That shouldindeed be so in isentropic processes. The only form of energy exchange is work:

dW

dt¼ tr

dH

dtqð0Þ

� �¼ ��h

2dxdt

sinhðb�hxð0Þ=2Þ: ð12:26Þ

In both the isothermal and the isentropic cases the power dW=dt is negative ifdx=dt is positive, i.e., an increasing magnetic field leads to work extraction fromthe qubit. (In the opposite case dx=dt\0; the magnetic field performs work on thequbit.)

Due to the extreme simple structure of the qubit, the thermodynamics of itsmagnetization is degenerate. In common macroscopic magnetism, the equilibriumenergy E and magnetization M are independent extensive variables. For the qubit,the average magnetization is defined as M ¼ 1

2 trðrzqÞ which leads to the followingconstraint between E and M:

E ¼ ��hxM: ð12:27Þ

Therefore it makes sense to construct the thermodynamic entropy S in function ofE or M but not in both. Apart from this, the thermodynamics of the qubit isperfectly meaningful.

12.7 Q-Carnot Cycle

Beside its historic role in heat-engine theory, the Carnot cycle is of extreme theo-retical value for thermodynamic reversibility. Heat Q [ 0 from a hot reservoir oftemperature Th can spontaneously go to a cold one of temperature Tc\Th but notvice versa. In thermodynamics, the transfer of heat Q from Th decreases the entropyby Q=Tc and the absorption of heat at Tc increases the entropy by Q=Tc [ Q=Th

hence the net entropy change is positive whereas the opposite transfer woulddecrease the net entropy. Thermodynamics formulates the impossibility of theopposite transfer by the second law: entropy never decreases in a closed system.

130 12 Qubit Thermodynamics

Carnot cycle is a smart open system that is able to transfer heat reversibly (i.e., atconstant entropy) between Th and Tc in both directions. For example, it extracts heatQh [ 0 from the hot reservoir and delivers heat jQcj ¼ jQh to the cold one. Thedifference L ¼ Qh � jQcj is turned into work performed by the machine on theenvironment. Here g ¼ L=Qh ¼ 1� j is the Carnot efficiency. If we re-supply thiswork into the machine, it will re-extract the heat from the cold reservoir, turn workinto heat, and return both of them to the hot reservoir.

The Carnot machine consists of a thermal equilibrium system, called theworking fluid, whose parameters are externally controlled. In our case, the workingfluid is a single qubit and we control whether it is in contact with the hot or thecold reservoir or is just isolated, and we control the external magnetic field x aswell.5 To pour heat from the hot reservoir into the cold one reversibly, the controlmeans the following four processes repeated cyclically:

1. Isothermal ‘‘compression’’: the qubit is in contact with the hot reservoir and weslowly decrease the magnetic field from x1 to x2\x1: The qubit state variesfrom qbh;x1

to qbh;x2:

2. Adiabatic ‘‘compression’’: the qubit is isolated, the magnetic field is furtherdecreased to jx2: The qubit state qbh;x2

is constant, but its temperaturebecomes Tc; hence we have to write it as qbc;jx2

:

3. Isothermal ‘‘expansion’’: the qubit is in contact with the cold reservoir and weslowly increase the magnetic field from jx2 to jx1: The qubit state varies fromqbc;jx2

to qbc;jx1:

4. Adiabatic ‘‘expansion’’: the qubit is isolated, the magnetic field is further increasedback to x1: The qubit state qbc;jx1

is constant, but its temperature becomes Th;

hence we have to write it as qbh;x1which restores the initial state of the cycle.

Following Sect. 12.6, we calculate the integral heat and work consumptions,respectively, in each step in turn:

1:Wh ¼1bh

cosh12bh�hx

� �jx1x2

[0; Qh ¼�h

2x sinh

12bh�hx

� �jx1x2�Wh[0

2:Whc ¼ gj�h

2x2 cosh

12bh�hx

� �jx2x1

\0; Qhc ¼ 0

3:Wc ¼1bc

cosh12bc�hx

� �jjx2jx1

\0; Qc ¼�h

2x sinh

12bc�hx

� �jjx2jx1�Wc\0

4:Wch ¼ gj�h

2x1 cosh

12bh�hx

� �jx1x2

[0; Qch ¼ 0:

In each cycle, the qubit absorbs the heat Qh [ 0 from the hot reservoir, delivers thework L ¼ �Wh �Whc �Wc �Wc [ 0 to the magnetic field, and loses the heatjQcj ¼ Qh � L to the cold reservoir. Running the cycle the other way around, these

5 cf. [5] by Geva and Kosloff.

12.7 Q-Carnot Cycle 131

quantities change sign and heat will be pumped from the cold to the hot reservoir atthe expense of work performed by the magnetic field on the qubit. Recalling thatj ¼ bh=bc ¼ 1� g; the above expressions confirm the Carnot efficiency:

L

Qh¼ g: ð12:28Þ

The Carnot cycle is absolutely significant for macroscopic thermodynamicreversibility: heat can be moved back-and-forth at constant thermodynamicentropy, all we need is external mechanical work. The Carnot cycle’s significancefor microscopic quantum reversibility is far from being fully understood.

12.8 Problems, Exercises

12.1 Terabyte equivalent to Joule/degree. Consider 1 g water at room temperatureand suppose we heat it up by 1�. Its informatic (von Neumann) entropyincreases by an incredible large number of bits. Let us calculate the order ofwater’s mass, necessarily a microscopic portion of 1 g, whose informaticentropy would increase just by one terabyte—a storage available foreveryone’s home computer now.

12.2 Landauer’s principle.6 The erasure of one bit information is accompanied bythe release of at least kBT ln 2 of heat. Let us argue for the principle. Method:consider a q-storage in thermal environment.

12.3 Universal thermalizer?. One would ask whether the Lindblad structure

C aqay � 12faya; qg

� �þ e�b�C ayqa� 1

2faay; qg

� �

on the r.h.s of Eq. (12.16) corresponds to a universal thermalizer, whether itmodels the influence of the thermal reservoir for any qubit? Suppose a dif-ferent Hamiltonian H ¼ �0n instead of H ¼ �n and find the stationary state.

12.4 Mechanical work on a qubit. Let us show, for pedagogical purposes, that thework performed by the variation of the external magnetic field x could beequivalently performed by an external mechanical force.

12.5 Adiabatic demagnetization. We apply a strong magnetic field x to our qubitat environmental temperature T and wait until the qubit becomes thermal-ized. Then we suddenly decrease the magnetic field to a much lower valuex0 � x: What happens to the temperature of the qubit?

6 Cf. [6] by Landauer.

132 12 Qubit Thermodynamics

References

1. Scarani, V., Ziman, M., Štelmanovic, P., Gisin, N., Buzek, V.: Phys. Rev. Lett. 88, 097905(2002)

2. Linden, N., Popescu, S., Skrzypczyk, P.: Phys. Rev. Lett. 105, 130401 (2010)3. Alicki, R.: J. Phys. 12, 103 (1979)4. Spohn, H., Lebowitz, J.L.: Adv. Chem. Phys. 38, 109 (1979)5. Geva, E., Kosloff, R.: J. Chem. Phys. 96, 3054 (1992)6. Landauer, R.: IBM J. Res. Dev. 5, 183 (1961)

References 133

Appendix

How reliable is the principle (cf. Sect. 12.3) that identifies informatic and ther-modynamic entropies? We present an example1 where this principle, applied toboth thermodynamic and q-informatic analysis of a simple model of irreversibility,will lead to an independent new—and exact—feature of q-entropy.

A.1 Introduction

Consider a thermal reservoir at (inverse) temperature b. Thermodynamicsdetermines its thermodynamic entropy Sth

R . If qR is the Gibbs canonical densitymatrix of the reservoir, we can calculate the informatic (von Neumann) entropy aswell. We can, in principle, prove that the thermodynamic entropy Sth

R of ourequilibrium reservoir is equal to the informatic entropy:

SRth ¼ SðqRÞ � �trðqR log qRÞ; ðA:1Þ

provided we take the so-called thermodynamic, i.e., infinite volume limit. (Units inAppendix are chosen such that kBln2 = 1, to count both the thermodynamic andinformatic entropies in bits.) What happens if we move the system out of itsequilibrium? Suppose we switch on a certain external macroscopic field for awhile, then we switch it off. This perturbation will always increase the energy ofthe reservoir, the field will always perform work W [ 0 on it. From thethermodynamic viewpont, part of this work W will be dissipated into heat in thereservoir. We are interested in those situations where the whole of W getsdissipated. Then the increase of the thermodynamic entropy of the reservoir is ofthe standard form:

DSRth ¼ bW [ 0: ðA:2Þ

1 Cf. [1] by Diósi, Feldmann and Kosloff.

135

According to the principle discussed in Sect. 12.3 we might expect that thisirreversible thermodynamic entropy production equals the change of the infor-matic entropy of the reservoir. If q0R stands for the state after the external per-turbation, we might expect the following equation to hold:

DSRth ¼ Sðq0RÞ � SðqRÞ: ðA:3Þ

But it cannot, since the external macroscopic field makes qR evolve unitarily,reversibly. The perturbed state is of the form:

q0R ¼ URqRUyR ; ðA:4Þ

with unitary UR; hence the r.h.s. of (A.3) is always zero. This is the notoriousconflict between the reversibility of the reservoir’s microscopic dynamics and themacroscopically observed irreversible dissipation. Resolutions of thiscontradiction can be based on a smart completion of the unitary evolution by asubsequent irreversible q-operation M; then we might expect that the modifiedequation holds:

DSRth ¼ SðMq0RÞ � SðqRÞ: ðA:5Þ

Unfortunately, typical dissipative systems are so complex that we cannot fit ourchoice ofM to satisfy this equation. Either the thermodynamic calculation of DSth

R

or the microscopic calculation of SðMq0RÞ or both are unavailable. Therefore wechoose a specific strategy.

We consider the simplest ever thermal reservoir, which is the ideal qubit gas ofSect. 12.2, and we consider the simplest external perturbation, which is the single-qubit unitary map. This way we reduce technical difficulties to the minimum, theterms of Eq. A.5 become tractable technically. The equation becomes a powerfulcriterion to single out the irreversible q-operation M:

A.2 The Reservoir, Collisions

Consider a single qubit in constant magnetic field, with a Hamiltonian H, andassume that it is initially in the Gibbs equilibrium state q ¼ qb (12.1) at the inversetemperature b:

q ¼ N e�bH : ðA:6Þ

Assume an arbitrary short pulse superposed on the constant magnetic field to causea ‘collision’ to the qubit state:

q! UqUy � r; ðA:7Þ

136 Appendix

where U is unitary since the dynamics of the qubit is Hamiltonian all along thepulse. Suppose our thermal reservoir is formed by the ideal gas ofn distinguishable qubits (molecules) in equilibrium:

qR ¼ q� q� . . .q � q�n; ðA:8Þ

with the total Hamiltonian

HR ¼ H � I�ðn�1Þ þ I � H � I�ðn�2Þ þ I�ðn�1Þ � H; ðA:9Þ

and its thermodynamic limit is n? ?. Let us assume that the qubits of thereservoir will reversibly collide with the pulses of the field according to theEq. A.7. Without restricting the generality, we can assume that the 1st qubitcollides first, the 2nd collides second, etc.:

q�n ! r� q�ðn�1Þ ! r� r� q�ðn�2Þ ! . . . ðA:10Þ

Since SðrÞ ¼ SðqÞ; the informatic entropy of the reservoir cannot change at all inthe above reversible collisions:

Sðq�nÞ ¼ Sðr� q�ðn�1ÞÞ ¼ Sðr� r� q�ðn�2ÞÞ ¼ . . . ¼ nSðqÞ ¼ nSðrÞ: ðA:11Þ

For simplicity, we consider the first collision:

q0R ¼ r� q�ðn�1Þ; ðA:12Þ

and implement our idea for it. We define the average work W of the field pulseperformed on the qubit:

W � trðHrÞ � trðHqÞ: ðA:13Þ

We express H ¼ ðI logN � log qÞ=b from (A.6), substitute it, and observe thattrðq log qÞ ¼ trðr log rÞ:

W ¼ trðr log rÞ � trðr log qÞb

� SðrkqÞb

; ðA:14Þ

We have recognized the appearance of the relative q-entropy (A.9) of the post- andpre-collision states. Since the qubit is part of a reservoir and the field interacts withmany qubits in succession then, thermodynamically, we postulate that the aboveenergy (the work by the pulses) is dissipated to the reservoir. Therefore, theaverage thermodynamic entropy production per collision is DSth

R = bW, whichusing (A.14), reads:

DSRth ¼ SðrkqÞ: ðA:15Þ

We see that DSthR is always positive since SðrkqÞ is always positive if r 6¼ q: The

above identity is part of standard statistical physics.

Appendix 137

Our target identity is different, it is just (A.5) expressing the identity of thethermodynamic and informatic entropy productions, respectively. Let us substitutethe microscopic expression (A.15) of DSth

R , the expression (A.8) of qR; and theexpression (A.12) of q0R: We get the following equation between various vonNeumann entropies:

SðrkqÞ ¼ SðMr� q�ðn�1ÞÞ � Sðr� q�ðn�1ÞÞ: ðA:16Þ

We expect this identity to hold in the thermodynamic limit n? ?. We have yet tospecify the irreversible q-operation M:

A.3 The Graceful Irreversible Operation

We must postulate an irreversible q-operation M which satisfies the constraint(A.16) while it does but gracefully modify the exact post-collision state r�q�ðn�1Þ: The map

r� q�ðn�1Þ ! M r� q�ðn�1Þ� �

ðA:17Þ

should obviously be symmetric for the permutation of the qubits. Single-qubitmaps can increase the informatic entropy but they are not likely to produce therequested value SðrkqÞ: HenceM should correlate the qubits. On the other hand,M should only smear out information whose loss is heuristically justifiable in areservoir like ours. It must not change the total energy or the total magnetization.Interestingly, there is a beautiful simple choice: let M be, in general, the totalsymmetrization over all permutations of the n qubits. In our case it means:

M r� q�ðn�1Þ� �

¼ r� q�ðn�1Þ þ q� r� q�ðn�2Þ þ � � � þ q�ðn�1Þ � rn

: ðA:18Þ

It is clear that this post-collision operation is irreversible and increases theinformatic entropy of the reservoir. It is still an open issue if it produces as muchentropy SðrkqÞ as expected thermodynamically.

A.4 New Math Conjecture on Relative Q-Entropy

Let us summarize what we have done so far. For an ideal qubit gas we consideredsingle qubit reversible perturbations by repeated external pulses. We postulatedthat the work by the pulses is totally dissipated in the macroscopic thermodynamicsense. This way we postulated the amount of thermodynamic entropy productionper pulses. To produce microscopic irreversibility within the model, we imposed agraceful irreversible operation which we think would produce as much von

138 Appendix

Neumann entropy as thermodynamics did. We found that this latter conditiondepends on the validity of the following mathematical conjecture:

limn¼1

SðMðr� q�ðn�1ÞÞÞ � Sðr� q�ðn�1ÞÞ� �

¼ SðrkqÞ; ðA:19Þ

where M is full symmetrization (A.18).The derivation of the conjecture involved numerous heuristic steps, our choice

of the irreversible operationM was purely intuitive, and the heart of the derivationwas the principle that thermodynamic and informatic entropies must coincide evenalong irreversible processes. So, the conjecture could easily have proven wrong.But it is correct [2], we got a new independent theorem and interpretation for therelative q-entropy.

References

1. Diósi, L., Feldmann, T., Kosloff, R.: Int. J. Quant. Inf. 4, 99 (2006)2. Csiszár, I., Hiai, F., Petz, D.: J. Math. Phys. 48, 092102 (2007)

Appendix 139

Solutions

Problems of Chap. 2

2.1 Mixture of pure states. It is easy to see that qðxÞ ¼ qð0Þdx0 þ qð1Þdx1 whichsuggests that the distribution of the mixing weights must coincide with thedistribution q. Accordingly, in the case of continuous phase space, we canchoose wð�xÞ ¼ qð�xÞ to mix the pure states dðx� �xÞ:

qðxÞ ¼Z

wð�xÞdðx� �xÞd�x:

2.2 Probabilistic or non-probabilistic mixing? Mixing n zeros and n ones isrealized by their random permutation that amounts to the following 2n-partitecomposite state:

qðx1; . . .; x2nÞ ¼dx10. . .dxn0dxnþ11. . .dx2n1 þ permutations of x1; . . .; x2n

# of permutations:

In case of the probabilistic mixing, however, we make 2n random decisions asto choose a zero or a one and then we mix the 2n elements. The probabilisticmixing amounts to q(x1)…q(x2n) which is obviously different from the abovecomposite state.

2.3 Classical separability. If we choose wð�xA;�xBÞ ¼ qABð�xA;�xBÞ and replacesummation over the weights by integration, it works:

qABðxA; xBÞ ¼Z

wð�xA;�xBÞdðxA � �xAÞdðxB � �xBÞd�xAd�xB:

Thus all qAB are mixtures of the uncorrelated pure states dðxA � �xAÞdðxB � �xBÞ:2.4 Decorrelating a single state? The map qAB? qAqB is nonlinear:

qABðxA; xBÞ �!Z

qABðxA; x0BÞdx0B

ZqABðx0A; xBÞdx0A:

Hence the map is not a real operation.

141

2.5 Decorrelating an ensemble. The collective state q�2nAB is granted to start with.

We interchange the first n and the second n subsystems A. Then we trace overthe second n composite systems AB. We get ðqAqBÞ�n . This can be verified if,e.g., we calculate the expectation values of observables like f ðxA1ÞgðxB1Þ, andf1ðxA1Þf2ðxA2Þg1ðxB1Þg2ðxB2Þ, etc.

2.6 Measurement and Bayes theorem. The Bayes theorem states that the a prioridistribution q(x) will be updated if we learn the value n of a variable whichwas a priori correlated with x. To calculate the updated conditionaldistribution qðxjnÞ from the a priori q(x), we need to know the a prioridistribution q(n) of n as well as its conditional distribution qðnjxÞ:

qðxjnÞ ¼ qðnjxÞqðxÞqðnÞ :

The quantities qðxjnÞ and q(n) correspond to qn(x) and pn, respectively, in thenotations of the measurement scheme of Sect. 2.4. The above Bayesian pre-diction becomes completely identical with the prediction qn(x) of the mea-surement (2.20) if we identify the measured effects by the Bayesianconditional distributions of n:

PnðxÞ ¼ qðnjxÞ:

2.7 Indirect measurement. Imagine a detector of discrete state space {n}. Let thecomposite state of the system and the detector be q(x, n) = q(x)Pn(x) whereq(x) stands for the reduced state of the system. Observe that Pn(x) becomesthe conditional state qðnjxÞ of the detector for the system being in the purestate x. Now we perform a projective measurement on the detector quantityn. Formally, the partition {Pm} of the detector state space must be defined asPmðnÞ ¼ dmn for all m. According to the rules of projective measurement, thestate change will be

qðx; nÞ ! qmðx; nÞ �1

pmdmnqðx; nÞ;

with probability pm ¼R

qðx;mÞdx: For the reduced state qðxÞ ¼P

m qðx;mÞ;the above projective measurement induces the desired non-projectivemeasurement of the effects {Pn(x)}.

Problems of Chap. 3

3.1 Bohr quantization of the harmonic oscillator. The sum of the kinetic andpotential energies yields the total energy E ¼ 1

2p2 þ 1

2x2q2 which is constant

during the motion. Therefore the phase space point (q, p) moves on the ellipseof surface 2pE/x. The surface plays a role in the Bohr–Sommerfeldq-condition because the contour-integral of pdq for one period is equal to

142 Solutions

the surface of the enclosed ellipse. Hence the canonical action takes the formE/x and we get

E=x ¼ �h nþ 12

� �:

3.2 The role of adiabatic invariants. The canonical actions Ik are adiabaticinvariants of the classical motion. This means that they remain approximatelyconstant against whatever large variations of the external parameters of theHamilton-function provided the variations are slow with respect to the motion.So, the canonical action I of the oscillator will be invariant againstthe variation of x in the Hamiltonian 1

2p2 þ 1

2x2ðtÞq2 provided j _xj � x2:

The q-condition remains satisfied with the same q-number n.3.3 Classical-like or q-like motion. The Bohr–Sommerfeld q-condition restricts

the continuum of classical motions to a discrete infinite sequence. For smallq-numbers this restriction is relevant since the allowed phase spacetrajectories are well separated. For large q-numbers, typically, the allowedtrajectories become quite dense in phase space and might fairly approximateany classical trajectory which does otherwise not satisfy the q-conditions.

Problems of Chap. 4

4.1 Decoherence-free projective measurement. Let us construct the spectralexpansion A ¼

Pk AkPk and the post-measurement state q0 ¼

Pk PkqPk. If

A; q� �

¼ 0 then q0 ¼ q since A; q� �

¼ 0 is equivalent with Pk; q� �

¼ 0 for all k.

To prove the inverse statement, that q0 ¼ q implies ½A; q� ¼ 0; we can simplywrite A; q

� �¼ A; q0� �

¼ 0:4.2 Mixing the eigenstates. Let us consider the spectral expansion of the matrix q:

q ¼X

k

qkPk:

If q is non-degenerate then the Pk’s correspond to the pure eigenstates of q andtheir mixture yields the state q if the corresponding eigenvalues make the mixingweights: wk = qk. In the general case, the spectral expansion implies the mixtureq ¼

Pk wkqk with wk = dkqk and qk ¼ Pk=dk where dk is the dimension of Pk:

4.3 Weak measurement of correlation. We shall need both spectraldecompositions A ¼

Pk AkPk and B ¼

Pk BkQk. After the unsharp

measurement of A on the state q; the post-measurement state becomes this:

q�A ¼1

p�A

ffiffiffiffiffiffiffiffiffiffi2pr2p exp �ð

�A� AÞ2

4r2

" #q exp �ð

�A� AÞ2

4r2

" #;

Solutions 143

where p�A is the probability of the measurement outcome �A: At the end of theday, we shall take the weak measurement limit r? ?. Before it, we considerthe projective measurement of B on the above post-measurement state,yielding the outcome Bk with probability pk ¼ trðQkq�AÞ: Then we can write

h�ABki ¼Z

p�A�AX

k

pkBkd�A ¼Z

p�A�Atr Bq�A

d�A:

We insert the expression of q�A; we get an integral on the r.h.s.:

1ffiffiffiffiffiffiffiffiffiffi2pr2p

Z�A exp �ð

�A� AÞ2

4r2

" #q exp �ð

�A� AÞ2

4r2

" #d�A;

which can be evaluated if we insert the spectral decomposition of A; in theweak measurement limit it reduces to 1

2fA; qg: Using this result, we obtain the

desired equation h�ABki ¼ 12trðfA; BgqÞ:

4.4 Separability of pure states. If jwABi ¼ jwAi � jwBi then the composite densitymatrix qAB is a single tensor product and it is trivially separable. The otherway around, when the pure state satisfies the separability condition (4.48)

jwABihwABj ¼X

k

wkqAk � qBk;

then it follows that the matrices on both sides have rank 1. Accordingly, ther.h.s. must be equivalent to the tensor product of rank 1 (i.e.: pure state)density matrices:

jwABihwABj ¼ jwAihwAj � jwBihwBj;

which implies the form jwABi ¼ jwAi � jwBi:4.5 Unitary cloning? Let us suppose that we have duplicated two states jwi andjw0i:

jwi � jw0i �! jwi � jwi; jw0i � jw0i �! jw0i � jw0i:

The inner product of the two initial composite states is hwjw0i while the inner

product of the two final composite states is hwjw0i2. Therefore the aboveprocess of state duplication cannot be unitary.

Problems of Chap. 5

5.1 Pure state fidelity from density matrices. Observe that hmjni2 equals the traceof the product jnihnj times jmihmj: Let us invoke the Pauli-representation of

144 Solutions

these two density matrices and evaluate the trace of their product:

jhmjnij2 ¼ trI þ nr

2I þmr

2

� �¼ 1þ nm

2;

which yields cos2ð#=2Þ:5.2 Unitary rotation for j"i �! j#i. Since j"i corresponds to the north pole andj#i corresponds to the south pole on the Bloch-sphere, we need a p-rotationaround, e.g., the x-axis. The rotation vector is a ¼ ðp; 0; 0Þ and thecorresponding unitary transformation becomes

UðaÞ � exp � i

2ar

� �¼ �irx:

We can check the result directly:

�irxj"i ¼ �i0 11 0

� �10

� �¼ �i

01

� �¼ �ij#i:

5.3 Density matrix eigenvalues and states in terms of polarization. Consider thedensity matrix 1

2ðI þ srÞ and find the spectral expansion of sr: We learned thatif s is a unit vector then srj"si ¼ j"si and srj#si ¼ �j#si: If s B 1, the twoeigenstates remain the same and we keep the simple notations j"si; j#si todenote qubits polarized along or, respectively, opposite to the directions. The eigenvalues will change trivially and we have srj"si ¼ sj"si andsrj#si ¼ �sj#si: Then we can summarize the eigenvalues and eigenstates ofthe density matrix in the following way:

I þ sr

2j"si ¼ 1þ s

2j"si; I þ sr

2j#si ¼ 1� s

2j#si:

5.4 Magnetic rotation for j"i �! j#i. We must implement a p-rotation of thepolarization vector and we can choose the rotation vector (p, 0, 0) whichmeans p-rotation around the x-axis. In magnetic field x; the polarizationvector s satisfies the classical equation of motion _s ¼ �x� s meaning thats will rotate around the direction x of the field at angular velocity x.Accordingly, we can choose the field to point along the x-axis:x ¼ ðx; 0; 0Þ. The rotation angle p is achieved if we switch on the fieldfor a period t = p/x.

5.5 Interrelated qubit physical quantities.

Pn þ P�n ¼ I

2Pn � rn ¼ 2P�n þ rn ¼ I

5.6 Mixing non-orthogonal polarizations. Since the qubit density matrix is a linearfunction of the polarization vector, mixing the density matrices meansaveraging their polarization vectors with the mixing weights. Therefore our

Solutions 145

mixture has the following polarization vector:

s ¼ 13� ð0; 0; 1Þ þ 2

3� ð1; 0; 0Þ ¼ ð1=3; 0; 2=3Þ:

Problems of Chap. 6

6.1 Universality of Hadamard and phase operations. The unitary rotations U ofthe qubit space are, apart from an irrelevant phase, equivalent to the spatialrotations of the corresponding Bloch sphere. This time the three Euler anglesw, h, / are the natural parameters. We can write the unitary rotations,corresponding to the spatial ones, into this form:

Uðw; h;/Þ ¼ exp � i

2wrz

� �exp � i

2hrx

� �exp � i

2/rz

� �:

The middle factor, too, becomes rotation around the z-axis if we sandwich itbetween two Hadamard operations because HrxH ¼ rz: We can thus expressthe r.h.s. in the desired form

Uðw; h;/Þ ¼ TðwÞHTðhÞHTð/Þ:

6.2 Statistical error of qubit determination. Out of N, we allocate Nx, Ny, Nz

qubits to estimate sx, sy, sz, respectively. We learned that the estimated valueof sx takes this form:

N"x � N#xN"x þ N#x

¼ 2N"xNx� 1;

because on a large statistics Nx ¼ N"x þ N#x the ratio N"x=Nx converges to theq-theoretical prediction p"x ¼ h"xjqj"xi � 1

2ð1þ sxÞ: The statistical error ofthe estimation takes the form 2DN"x=Nx and we are going to determine themean fluctuation DN"x. The statistical distribution of the count N"x is binomial:

pðN"xÞ ¼Nx

N"x

� �pN"x"x pN#x

#x ;

hence the mean squared fluctuation of the count N"x takes the form

ðDN"xÞ2 ¼ Nxp"xp#x ¼ Nxð1� s2xÞ=4. This yields the ultimate form of the

estimation error:

Dsx ¼

ffiffiffiffiffiffiffiffiffiffiffiffi1� s2

x

Nx

s;

and we could get similar results for Dsy and Dsz.

146 Solutions

6.3 Fidelity of qubit determination. If the state jni sent by Alice and thepolarization rm chosen by Bob were fixed then the structure of the expectedfidelity of Bob’s guess would be this:

jhnjmij2p"m þ jhnj�mij2p#m:

Here we have understood that Bob’s optimum guess must always be the post-measurement state jmi based on the measurement outcome rm ¼ 1;

respectively. Now we recall that jhnjmij2 ¼ p"m ¼ cos2ð#=2Þ where cos# ¼nm; and jhnj�mij2 ¼ p#m ¼ sin2ð#=2Þ: Hence the above fidelity takes thesimple form cos4ð#=2Þ þ sin4ð#=2Þ which we rewrite into the equivalent form12þ 1

2 cos2ð#Þ: The average of cos2ð#Þ ¼ ðnmÞ2 over the random independentn and m yields 1/3 therefore the expected fidelity of Bob’s guess becomes 2/3.

6.4 Post-measurement depolarization. Let rn denote the polarization chosen by Bob.The non-selective measurement induces the change q! PnqPn þ P�nqP�n ofthe state. Inserting the Pauli-representation of q and the projectors Pn yields

I þ sr

2! I þ rn

2I þ sr

2I þ rn

2þ I � rn

2I þ sr

2I � rn

2

¼ I þ sr

4þ I þ srnrrn

4:

Since Bob’s choice is random regarding n we shall average n over the solidangle. Averaging the structure rnrrn yields �r=3 hence the average influenceof Bob’s non-selective measurements can be summarized as

I þ sr

2! I þ sr=3

2:

6.5 Anti-linearity of polarization reflection. Let us calculate the influence of theanti-unitary transformation T on a pure state qubit:

Tjni ¼ T cosh2j"i þ eiu sin

h2j#i

� �¼ � cos

h2j#i þ e�iu sin

h2j"i ¼ e�iuj�ni:

6.6 General qubit effects. We can suppose that the weights wn are non-vanishing. First,we have to impose the conditions janj 1 since otherwise the matrices would beindefinite. Second, the request Pn� 0 implies the conditions wn [ 0. And third,the request

Pn Pn ¼ I implies the conditions

Pn wn ¼ 1 and

Pn wnan ¼ 0.

Problems of Chap. 7

7.1 Schmidt orthogonalization theorem. Let r be the rank of c and let us consider

the non-negative matrices ccy and cyc of rank r both. Their spectrum is

Solutions 147

non-negative and identical. Indeed, if cycjRi ¼ wjRi; i.e., w and |Ri are an

eigenvalue and a (normalized) eigenvector of cyc then jLi ¼ cjRi=ffiffiffiffiwp

will be

a (normalized) eigenvector of ccy with the same eigenvalue w. This can beseen by direct inspection. Now we determine the r positive eigenvalues wk for

k = 1, 2, …, r and the corresponding orthonormal eigenstates jk; Ri of cyc.

Then, by jk; Li ¼ cjk; Ri= ffiffiffiffiffiffiwkp

; we define the r orthonormal eigenstates of ccywhich belong to the common positive eigenvalues wk, fork = 1, 2, …, r. Now we can see that

cjk; Ri ¼ ffiffiffiffiffiffiwkp jk; Li;

for all k = 1, 2, …, r. We have thus proven that there exists the followingSchmidt decomposition of the matrix c:

c ¼Xr

k¼1

ffiffiffiffiffiffiwkp jk; Lihk; Rj:

7.2 Swap operation. For convenience, we use r ¼ ðrx iryÞ=2 instead of rx; ry:

Now we express the Pauli matrices in the up-down basis:

rþ ¼ j"ih#j; r� ¼ j#ih"j; rz ¼ j"ih"j � j#ih#j:

Substituting these expressions we obtain

I � I þ r� r

2¼ I � I þ 2rþ � r� þ 2r� � rþ þ rz � rz

2¼ j""ih""j þ j##ih##j þ j"#ih#"j þ j#"ih"#j;

which is indeed the swap matrix S:7.3 Singlet density matrix. The singlet state qðsingletÞ is invariant under rotations

of the Bloch sphere. Therefore qðsingletÞ must be of the form

qðsingletÞ ¼ I � I þ kr� r

4

because there are no further rotational invariant mathematical structures. We

could determine the parameter k from the pure state condition ½qðsingletÞ�2 ¼qðsingletÞ; yielding k = -1. However, we can spare these calculations if werecall the swap S. It is Hermitian, rotation invariant and idempotent: S2 ¼I � I: Hence we get the singlet state directly in the form

qðsingletÞ ¼ I � I � S

2¼ I � I � r� r

4:

7.4 Local measurement of expectation values. Alice and Bob will determine hA�Bi and hA0 � B0i separately on two independent sub-ensembles and will finally

148 Solutions

add them since the expectation value is additive. Still we have to show that theexpectation value of a tensor product, like hA� Bi, can be determined in localmeasurements. We introduce the local spectral expansions A ¼

Pk AkPk and

B ¼P

l BlQl. Alice and Bob perform local measurements of A and B incoincidence, yielding the measurement outcomes A1, B1, A2, B2, …, Ar,Br, …, AN, BN where Ar is always an eigenvalue Ak and the case is similarfor the Br’s. Then Alice and Bob can calculate the q-expectation valueasymptotically:

hA� Bi ¼ limN!1

1N

Xr

ArBr:

To see that this is indeed the right expression of hA� Bi we have to rethinkthe nonlocal measurement of A� B itself. Its spectral expansion is

A� B ¼Xðk;lÞðAkBlÞðPk � QlÞ;

and the corresponding q-measurement will obviously yield the same statisticsof the outcomes ArBr like in case of the local-measurements.

7.5 Local measurement of certain nonlocal quantities. If we measure rz � rz on asinglet state we always get -1 and the singlet state remains the post-measurement state. In the attempted local measurement, the entanglement isalways destroyed and we get either j"#i or j#"i for the post-measurement state.Obviously the degenerate spectrum of rz � rz plays a role in the nonlocality. If,in the general case, we suppose A� B has a non-degenerate spectrum then thepost-measurement states will be the same pure states in both the nonlocalmeasurement of A� B and the simultaneous local measurements of A and B.

7.6 Nonlocal hidden parameters. Let the further hidden parameter m take values1, 2, 3, 4 marking whether Alice and Bob measures A� B; A0 � B; A� B0 orA0 � B0; respectively. Then, according to the hidden variable concept, theassignment of all four polarization values will uniquely depend on thecomposite hidden variable rm:

A ¼ Arm ¼ 1; A0 ¼ A0rm ¼ 1; B ¼ Brm ¼ 1; B0 ¼ B0rm ¼ 1:

Contrary to the local assignment (7.36), the above assignments are callednonlocal since the hidden variable rm is nonlocal: it depends on both Alice’sand Bob’s measurement setup. The statistical relationships, cf. (7.37), becomemodified:

hA� Bi ¼ limN1!1

1N1

Xr2X1

Ar1Br1; N1 ¼ jX1j;

hA0 � Bi ¼ limN2!1

1N2

Xr2X2

Ar2Br2; N2 ¼ jX2j;

Solutions 149

etc. for the other two cases m = 3, 4. The assignments are independent for thefour different values of m. There is no constraint combining the Arm’s withdifferent m’s. Hence it has become straightforward to reproduce the above said

four q-theoretical predictions including of course correlations hCi that arehigher than 2.

7.7 Does teleportation clone the qubit? The selective post-measurement state ofthe two qubits on Alice’s side is one of the four maximally entangledBell-states. Therefore the reduced state of the qubit that she had teleported isleft in the totally mixed state independently of its original form as well as ofthe four outcomes of Alice’s measurement. Note that the form (7.47) of thethree-qubit pre-measurement state shows that Alice’s measurement outcome isalways random. The four outcomes have probability 1/4 each.

Problems of Chap. 8

8.1 All q-operations are reductions of unitary dynamics. Given the trace-

preserving q-operation Mq ¼P

n MnqMyn ; we have to construct the unitary

interaction matrix U acting on the composite state of the system andenvironment. Let us introduce the composite basis jki � jn; Ei wherek = 1, 2, …., d and n = 1, 2, …, dE. Let us define the influence of U on asubset of the composite basis:

U jki � j1; Eið Þ ¼XdE

n¼1

Mnjki � jn; Ei; k ¼ 1; 2; . . .; d:

This definition is possible because the above map generates orthonormalvectors:

XdE

m¼1

hljMym � hm; EjXdE

n¼1

Mnjki � jn; Ei ¼XdE

n¼1

hljMyn Mnjki ¼ dkl:

The further matrix elements of U; i.e. those not defined by our first equationabove, can be chosen in such a way that U is unitary on the whole compositestate. Using this definition of U in the equation (8.3) of reduced dynamics we

can directly inspect that the resulting operation is Mq ¼P

n MnqMyn ; as

expected.8.2 Non-projective effect as averaged projection. Let us substitute the proposed

form of the effects Pn into the equation pn ¼ trðPnqÞ introduced for non-projective measurement in Sect. 4.4.2

150 Solutions

tr Pnq

¼ tr trEPnqEq

¼ tr PnqEq

:

In this formalism, i.e., without the �’s, the matrices of different subsystemscommute hence qEq ¼ qqE: Thus we obtain the following result: trðPnqÞ ¼trðPnqqEÞ: We recognize the coincidence of the r.h.s. with the r.h.s. of (8.18).Since this coincidence is valid for all possible q; it verifies the proposed formof Pn:

8.3 Q-operation as supermatrix. We start from the Kraus representation Mq ¼Pn MnqM

yn : We take the matrix elements of both sides and we also sandwich

the q between the identitiesP

k0 jk0ihk0j and

Pl0 jl0ihl0j on the r.h.s.:

hkjMqjli ¼ hkjX

n

Mn

Xk0jk0ihk0jq

Xl0jl0ihl0jMyn jli:

Comparing the r.h.s. withP

k0l0 Mklk0l0qk0l0 , we read out the components of

the supermatrix: Mklk0l0 ¼P

nhkjMnjk0ihl0jMyn jli:

8.4 Environmental decoherence, time-continuous depolarization. The equationtakes the Lindblad form with H ¼ 0 and with three hermitian Lindbladmatrices identified by the Cartesian components of ðr=2

ffiffiffispÞ: For

convenience, we shall use the Einstein convention to sum over repeatedindices, e.g.: sr ¼ sara: We write the r.h.s. of the master equation into theequivalent form �ð1=8sÞ rb; rb; q½ �½ � and insert q ¼ 1

2ðI þ saraÞ into it. Themaster equation reduces to

_sara ¼ �18s

rb; rb; sara½ �½ � ¼ �1ssara;

which means the simple equation _s ¼ �s=s for the polarization vector. Itssolution is sðtÞ ¼ e�t=ssð0Þ: Therefore the s may be called depolarization time,or decoherence time as well.

8.5 Kraus representation of depolarization. The map should be of the formMq ¼ð1� 3kÞqþ krqr with 0 B k B 1/3 since there exist no other isotropic Krausstructures for a qubit. The depolarization channel decreases the polarizationvector s by a factor 1 - w and we have to find the parameter k as function ofw. Inserting q ¼ 1

2ðI þ srÞ we get

MI þ sr

2¼ I þ ð1� 4kÞsr

2;

which means that k = w/4. Four Kraus matrices make the depolarization

channel:ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 3w=4

pI and the three components of

ffiffiffiffiffiffiffiffiffiw=4

pr:

Solutions 151

Problems of Chap. 9

9.1 Positivity of relative entropy. We can write

Sðqkq0Þ ¼X

x

qðxÞ logqðxÞq0ðxÞ:

We invoke the inequality ln k[ 1 - k-1 valid for k = 1 and apply it tok = q/q0. This yields

Sðqkq0Þ[ 1ln 2

Xx

qðxÞ 1� q0ðxÞqðxÞ

� �¼ 0;

which always holds if q0 = q.9.2 Concavity of entropy. Suppose we have a long message x1

(1)x2(1)…xn

(1) whereq1(x) is the a priori distribution of one letter. Let S1 stand for the single-letterentropy S(q1). The number of the typical messages is 2nS1 so that their shortestcode is nS1 bits. Consider a second message from the same alphabet andsuppose the single-letter distribution q2(x) is different from q1(x). Let usconcatenate the two messages:

xð1Þ1 xð1Þ2 xð1Þ3 . . .xð1Þn xð2Þ1 xð2Þ2 xð2Þ3 . . .xð2Þm ;

where the two lengths n and m may be different. Obviously, the number of thetypical ones among such composite messages is 2nS1 � 2mS2 and their shortestcode is nS1 + mS2 bits. Now imagine that we permute the n + m lettersrandomly. On one hand, the composite messages become usual (n + m)-letter-long messages where the single letter distribution is always the same, i.e., themixture q = w1q1 + w2q2 with weights w1 = n/(n + m) and w2 = m/(n + m).Therefore the number of the typical messages is 2(n+m)S(q) and the shortest codeis (n + m)S(q) bits. On the other hand, we can inspect that the number of thetypical messages 2(n+m)S(q) is greater than 2nS1 � 2mS2 because the number ofinequivalent permutations has increased: the first n letters have becomepermutable with the last m letters. This means that (n + m)S(q) [ nS1 + mS2

which is just the concavity of the entropy: S(w1q1 + w2q2) [ w1S(q1) +w2S(q2), for q1 = q2.

9.3 Subadditivity of entropy. We can make the choice q0AB (x, y) = qA(x)qB(y). Tocalculate SðqABkq0ABÞ ¼ �SðqABÞ �

Px;y qABðx; yÞ log q0ABðx; yÞ, we note that

the second term is �P

x;y qABðx; yÞ log qAðxÞqBðyÞ½ � ¼ SðqAÞ þ SðqBÞ: Hence

the positivity of the relative entropy SðqABkq0ABÞ� 0 proves subadditivity:S(qAB) B S(qA) + S(qB).

9.4 Coarse graining increases entropy. Let us identify our system by the k-partitecomposite system of the k bits x1, x2, …, xk. Then the coarse grained systemcorresponds to the (k - 1)-partite sub-system consisting of the first k - 1 bitsx1, x2,…, xk-1. The coarse grained state ~q is just a reduced state w.r.t. q. Hencewe see that coarse graining increases the entropy because reduction does it.

152 Solutions

Problems of Chap. 10

10.1 Subadditivity of q-entropy. Let us calculate

SðqABkqA � qBÞ ¼ �SðqABÞ � tr qAB logðqA � qBÞ½ �

and note that the second term is SðqAÞ þ SðqBÞ: Hence the Klein inequalitySðq0kqÞ� 0 proves the subadditivity: SðqABÞ SðqAÞ þ SðqBÞ:

10.2 Concavity of q-entropy, Holevo entropy. We assume a certain environmentalsystem E and a basis {|n; Ei} for it. Let us construct a composite state

qbig ¼X

n

wnqn � jn; Eihn; Ej:

Note that the reduced state of the system is invariably q ¼P

n wnqn and thereduced state of the environment is qE ¼

Pn wnjn; Eihn; Ej: Subadditivity

guarantees that SðqbigÞ SðqÞ þ SðqEÞ. Let us calculate and insert theentropies SðqbigÞ ¼ SðwÞ þ

Pn wnSðqnÞ and SðqEÞ ¼ SðwÞ which results in

the desired inequality:P

n wnSðqnÞ SðqÞ:

10.3 Data compression of the non-orthogonal code. The density matrix of thecorresponding 1-letter q-message reads

q ¼ j"zih"zj þ j"xih"xj2

¼ I þ rn=ffiffiffi2p

2;

which is a partially polarized state along the skew direction n ¼ ð1; 0; 1Þ=ffiffiffi2p

;cf. (6.29). The eigenvalues of this density matrix are the following:

pþ ¼1þ 1=

ffiffiffi2p

2; p� ¼

1� 1=ffiffiffi2p

2;

hence its von Neumann entropy amounts to

SðqÞ ¼ �pþ log pþ � p� log p� � 0:60:

According to the q-data compression theorem, we can compress one qubit ofthe q-message into 0.6 qubit on average, and this is the best maximumfaithful compression.

10.4 Distinguishing two non-orthogonal qubits: various aspects. In fact, wemeasure the polarization component orthogonal to the polarization of thesingle-letter density matrix. The measurement outcomes ±1 on both q-statesj"zi; j"xi will appear with probabilities p+ and p- (cf. Prob. 10.3), inalternating order of course:

Solutions 153

pðy ¼ þ1jx ¼ 0Þ ¼ pþ; pðy ¼ �1jx ¼ 0Þ ¼ p�pðy ¼ þ1jx ¼ 1Þ ¼ p�; pðy ¼ �1jx ¼ 1Þ ¼ pþ:

Regarding the randomness of the input message the output message, too,becomes random: H(Y) = 1. Hence the information gain takes this form andvalue:

Igain ¼ HðYÞ � HðY jXÞ ¼ 1þ pþ log pþ þ p� log p� � 0:40:

10.5 Simple optimum q-code. The q-data compression theory says that a pure stateq-code is not compressible faithfully (i.e.: allowing the same accessibleinformation) if and only if the single-letter average state has the maximumvon Neumann entropy. In our case, we must assure the following:

jRihRj þ jGihGj þ jBihBj3

¼ I

2;

which is possible if we chose three points on a main circle of the Bloch-sphere, at equal distances from each other.

Problems of Chap. 11

11.1 Creating the totally symmetric state.

jSi � 1

2n=2

X2n�1

x¼1

jx1x2. . .xni ¼X1

x1¼0

X1

x2¼0

. . .X1

xn¼0

jx1i � jx2i � . . .� jxni

¼ Hj0i � Hj0i � . . .� Hj0i � H�nj0i�n:

11.2 Constructing Z-gate from X-gate.

HXH ¼ 1ffiffiffi2p 1 1

1 �1

� �0 11 0

� �1ffiffiffi2p 1 1

1 �1

� �¼ 1 0

0 �1

� �¼ Z:

The inverse relationship follows from H2 ¼ I:

11.3 Constructing controlled Z-gate from cNOT-gate.

154 Solutions

11.4 Constructing controlled phase-gate from two cNOT-gates.

11.5 Q-circuit to produce Bell states. Let us calculate the successive actions of theH-gate and the cNOT-gate:

j00i �! j00i þ j10i �! j00i þ j11i �! jUðþÞij01i �! j01i þ j11i �! j01i þ j10i �! jWðþÞij10i �! j00i � j10i �! j00i � j11i �! jUð�Þij11i �! j01i � j11i �! j01i � j10i �! jWð�Þi

The trivial factors 1=ffiffiffi2p

in front of the intermediate states have not beendenoted.

11.6 Q-circuit to measure Bell states. The task is the inverse task of preparing theBell states. Since both the H-gate and the cNOT-gate are the inverses ofthemselves, respectively, we can simply use them in the reversed order w.r.t.the circuit that prepared the Bell states (cf. Prob. 11.5):

The boxes M stand for projective measurement of the computational basis.

Problems of Chap. 12

12.1 Terabyte equivalent to Joule/degree. To heat up 1 g water by 1 K we mustdeliver 4.186 J heat to it. If we adopt 300 K for room temperature, theincrease of the thermodynamic entropy will be Sth = (4.186/300) J/K. UsingkB = 1.381 9 10-23 J/K, our Eq. (12.11) yields S*1021 bits for the increaseof the informatic entropy. One terabyte is just 8 9 1012 bits, it corresponds toan amount of water less than 1 g by cca. 9 orders of magnitude. We concludethat the rate 1 terabyte/degree of informatic entropy increase would corre-spond to 1 ng water—a drop of size * 10 lm.

12.2 Landauer’s principle. Let us consider a large q-storage consisting of N qubitsand suppose, for concreteness, that they contain maximally compressed data.The von Neumann entropy is S = N. Suppose that the environment of the q-

Solutions 155

storage is thermal, its temperature is T, and it is uncorrelated with theq-storage. When the storage becomes erased, i.e., all qubits are set to j0i, thevon Neumann entropy drops to S = 0. If the erasure has been done reversiblyfrom the viewpont of the total (storage + environment) system then the totalq-entropy must not change. If the storage and the environment areuncorrelated after the erasure, too, then the change -N of the q-storageentropy will be compensated by the increase NkBln2 of the thermodynamicentropy of the environment. This is achieved by dissipating work NkBTln2into heat in the environment. The dissipated heat is, indeed, kBTln2 per oneerased qubit. If the erasure were not reversible, the released heat would begreater.

12.3 Universal thermalizer? To solve Eq. (12.16) with the Hamiltonian H0 ¼ �0n;we use the common interaction picture of quantum theory. This means thereplacements a! expð�i�0t=�hÞa and qt ! expði�0nt=�hÞqt expð�i�0nt=�hÞ.The master equation becomes

dqdt¼ C aqta

y � 12faya; qtg

� �þ e�b�C ayqta�

12faay; qtg

� �:

The self-Hamiltonian term cancels—this is always so in interaction picture.

Also the time dependence of a and ay cancel—this is a special consequenceof the Lindblad structure. Note that the above obtained form is not sensitiveto the value of the qubit energy gap �0. Its stationary solution will invariablybe qb with the parameter � which would correspond to a Gibbs state attemperature different from T by a factor �0=�. This incapacity of the masterequation follows from the underlying microscopic model. The thermalreservoir q�N

b thermalizes to the same (inverse) temperature b provided the

qubit to be thermalized has exactly the same Hamiltonian as the reservoir’squbits have. This is not the case here.

12.4 Mechanical work on a qubit. Suppose the vertical magnetic field x depends onthe z-coordinate. The mean energy E ¼ trðHqÞ of the isolated qubit behaveslike a z-dependent potential energy. To keep the qubit at a certain verticallocation z one must exert a mechanical force dE/dz. Now, one starts to vary theexternal magnetic field just by moving the qubit along the vertical axis. Thisrequires mechanical work W against the magnetic field gradient. The requestedpower is ðdE=dzÞðdz=dtÞ ¼ tr ðdH=dzÞq

� �ðdz=dtÞ ¼ tr ðdH=dtÞq

� �which

coincides with the definition (12.24) of the power dW/dt.12.5 Adiabatic demagnetization. The thermalized state of the qubit in the field x

is qb;x ¼ N expðb�hxrzÞ: If we change the field quickly to x0, the stateremains as it was. The thermal environment will drive it to the new thermalstate qb;x0 ¼ N expðb�hx0rzÞ: But it takes time and right after the quicktransition x ? x0 the qubit is surely of the previous equilibrium state qb;x.

156 Solutions

Now we observe that this state, at the current (smaller) field, corresponds toan other thermal state

qb;x ¼ N expðb�hxrzÞ ¼ N expðb0�hx0rzÞ ¼ qb0;x0

where b0 ¼ ðx=x0Þb. Hence the effective temperature T0 = b0/kB of our qubit is

T 0 ¼ x0

xT � T :

This means that, during the process of re-thermalization to temperature T, ourqubit absorbes heat from the environment, it acts as a refrigerator.

Solutions 157

Index

AAlice, Bob, 41, 51, 63, 66, 67,

69, 71, 72Alice, Bob, Eve, 55

BBayes theorem, 8Bell

basis, states, 65inequality, 70nonlocality, 68

bit, 13

Cchannel capacity, 91code, 88, 123

optimum, 92q-, 96superdense, 71

completely positive map, 23, 77contrary to classical, 22, 23, 26, 34

Ddata compression, 88, 97, 123decoherence, 26density matrix, 22, 38

Eentanglement, 21, 82

as resource, 100dilution, 102distillation, 64, 101maximum, 65

measure, 61two-qubit, 64

entropyconditional, 91

informatic, 123, 135relative, 88, 96, 138Shannon, 62, 87thermodynamic, 123, 135von Neumann, 62, 95, 135

equationmaster, 29, 83, 125of motion, 5, 21, 39

expectation value, 9, 27

Ffidelity, 40, 51Fock representation, 42, 124function evaluation, 108

Iinformation,

q-information, 13, 95accessible, 99mutual, 90theory, 87, 95

irreversiblemaster equation, 29operation, 12, 33, 47, 138q-measurement, 26reduced dynamics, 83

KKlein inequality, 76Kraus form, 77

159

LLindblad form, 83, 125local

Hamilton, 63operation, 63physical quantity, 63

LOCC, 82

Mmeasurement, 8, 24, 40

continuous, 10, 27in pure state, 30indirect, 79, 81non-projective, 10, 21, 27, 52, 81non-selective, 10, 26projective, 8, 25, 52selective, 9, 25unsharp, 10weak, 10, 27

message, 88, 96typical, 89

mixing, 6, 22, 48, 49

Nnonlocality

Bell, 68Einstein, 66

Ooperation, 6, 22, 77

depolarization, 47irreversiblelocal, 63logical, 63non-selective, 4, 23one-qubit, 45reflection, 47selective, 7, 23

PPauli, 35

matrices, 37representation, 35, 41

physical quantities, 8, 24compatible, 29

Qq-algorithm

error correction, 116

Fourier, 113oracle problem, 109period finding, 114searching, 111

q-banknote, 54q-Carnot cycle, 128q-channel, 56, 83q-circuit, 118q-computation, 103

parallel, 103representation

q-correlation, 71, 32history, 66

q-cryptography, 55q-entropy, 95q-gate, 118

universal, 47q-information

hiddenq-key, 55q-protocol, 54q-refrigerator, 125q-state

cloning, 51determination, 48, 51indistinguishability, 51, 54no-cloning, 50, 54non-orthogonal, 51preparation, 48purificationunknown, 45, 49

q-thermalizationqubit, 18, 35

external work, 127ideal gas, 122thermal, 121unknown, 41

Rreduced dynamics, 12, 78reservoirrotational invariance

SSchmidt decompositionselection, 6, 22state space, 5, 21

discrete, 14state, q-state

mixed, 5, 21pure, 5, 21separable, 12

160 Index

superposition, 18, 21system

bipartite, 59collective, 13composite, 12environmental, 60open, 86

Tteleportation, 72

Uurn model, 5–7, 22, 24

Index 161